<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <title>Numerical calculus — mpmath v0.17 documentation</title> <link rel="stylesheet" href="../_static/default.css" type="text/css" /> <link rel="stylesheet" href="../_static/pygments.css" type="text/css" /> <script type="text/javascript"> var DOCUMENTATION_OPTIONS = { URL_ROOT: '../', VERSION: '0.17', COLLAPSE_MODINDEX: false, FILE_SUFFIX: '.html', HAS_SOURCE: true }; </script> <script type="text/javascript" src="../_static/jquery.js"></script> <script type="text/javascript" src="../_static/doctools.js"></script> <link rel="top" title="mpmath v0.17 documentation" href="../index.html" /> <link rel="next" title="Polynomials" href="polynomials.html" /> <link rel="prev" title="q-functions" href="../functions/qfunctions.html" /> </head> <body> <div 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href="polynomials.html#polynomial-evaluation-polyval">Polynomial evaluation (<tt class="docutils literal"><span class="pre">polyval</span></tt>)</a></li> <li class="toctree-l2"><a class="reference external" href="polynomials.html#polynomial-roots-polyroots">Polynomial roots (<tt class="docutils literal"><span class="pre">polyroots</span></tt>)</a></li> </ul> </li> <li class="toctree-l1"><a class="reference external" href="optimization.html">Root-finding and optimization</a><ul> <li class="toctree-l2"><a class="reference external" href="optimization.html#root-finding-findroot">Root-finding (<tt class="docutils literal"><span class="pre">findroot</span></tt>)</a></li> </ul> </li> <li class="toctree-l1"><a class="reference external" href="sums_limits.html">Sums, products, limits and extrapolation</a><ul> <li class="toctree-l2"><a class="reference external" href="sums_limits.html#summation">Summation</a></li> <li class="toctree-l2"><a class="reference external" href="sums_limits.html#products">Products</a></li> <li class="toctree-l2"><a class="reference external" href="sums_limits.html#limits-limit">Limits (<tt class="docutils literal"><span class="pre">limit</span></tt>)</a></li> <li class="toctree-l2"><a class="reference external" href="sums_limits.html#extrapolation">Extrapolation</a></li> </ul> </li> <li class="toctree-l1"><a class="reference external" href="differentiation.html">Differentiation</a><ul> <li class="toctree-l2"><a class="reference external" href="differentiation.html#numerical-derivatives-diff-diffs">Numerical derivatives (<tt class="docutils literal"><span class="pre">diff</span></tt>, <tt class="docutils literal"><span class="pre">diffs</span></tt>)</a></li> <li class="toctree-l2"><a class="reference external" href="differentiation.html#composition-of-derivatives-diffs-prod-diffs-exp">Composition of derivatives (<tt class="docutils literal"><span class="pre">diffs_prod</span></tt>, <tt class="docutils literal"><span class="pre">diffs_exp</span></tt>)</a></li> <li class="toctree-l2"><a class="reference external" href="differentiation.html#fractional-derivatives-differintegration-differint">Fractional derivatives / differintegration (<tt class="docutils literal"><span class="pre">differint</span></tt>)</a></li> </ul> </li> <li class="toctree-l1"><a class="reference external" href="integration.html">Numerical integration (quadrature)</a><ul> <li class="toctree-l2"><a class="reference external" href="integration.html#standard-quadrature-quad">Standard quadrature (<tt class="docutils literal"><span class="pre">quad</span></tt>)</a></li> <li class="toctree-l2"><a class="reference external" href="integration.html#oscillatory-quadrature-quadosc">Oscillatory quadrature (<tt class="docutils literal"><span class="pre">quadosc</span></tt>)</a></li> <li class="toctree-l2"><a class="reference external" href="integration.html#quadrature-rules">Quadrature rules</a></li> </ul> </li> <li class="toctree-l1"><a class="reference external" href="odes.html">Ordinary differential equations</a><ul> <li class="toctree-l2"><a class="reference external" href="odes.html#solving-the-ode-initial-value-problem-odefun">Solving the ODE initial value problem (<tt class="docutils literal"><span class="pre">odefun</span></tt>)</a></li> </ul> </li> <li class="toctree-l1"><a class="reference external" href="approximation.html">Function approximation</a><ul> <li class="toctree-l2"><a class="reference external" href="approximation.html#taylor-series-taylor">Taylor series (<tt class="docutils literal"><span class="pre">taylor</span></tt>)</a></li> <li class="toctree-l2"><a class="reference external" href="approximation.html#pade-approximation-pade">Pade approximation (<tt class="docutils literal"><span class="pre">pade</span></tt>)</a></li> <li class="toctree-l2"><a class="reference external" href="approximation.html#chebyshev-approximation-chebyfit">Chebyshev approximation (<tt class="docutils literal"><span class="pre">chebyfit</span></tt>)</a></li> <li class="toctree-l2"><a class="reference external" href="approximation.html#fourier-series-fourier-fourierval">Fourier series (<tt class="docutils literal"><span class="pre">fourier</span></tt>, <tt class="docutils literal"><span class="pre">fourierval</span></tt>)</a></li> </ul> </li> </ul> </div> </div> </div> </div> <div class="sphinxsidebar"> <div class="sphinxsidebarwrapper"> <h4>Previous topic</h4> <p class="topless"><a href="../functions/qfunctions.html" title="previous chapter">q-functions</a></p> <h4>Next topic</h4> <p class="topless"><a href="polynomials.html" title="next chapter">Polynomials</a></p> <h3>This Page</h3> <ul class="this-page-menu"> <li><a href="../_sources/calculus/index.txt" rel="nofollow">Show Source</a></li> </ul> <div id="searchbox" style="display: none"> <h3>Quick search</h3> <form class="search" action="../search.html" method="get"> <input type="text" name="q" size="18" /> <input type="submit" value="Go" /> 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