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<div class="chapter">
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<div class="titlepage"><div><div><h1 class="title">
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<a name="root_finding"></a>Chapter 10. Root Finding & Minimization Algorithms</h1></div></div></div>
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<div class="toc">
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<p><b>Table of Contents</b></p>
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<dl class="toc">
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<dt><span class="section"><a href="math_toolkit/roots_noderiv.html">Root Finding Without Derivatives</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/roots_noderiv/bisect.html">Bisection</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/roots_noderiv/bracket_solve.html">Bracket and
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Solve Root</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/roots_noderiv/TOMS748.html">Algorithm TOMS 748:
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Alefeld, Potra and Shi: Enclosing zeros of continuous functions</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/roots_noderiv/brent.html">Brent-Decker Algorithm</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/roots_noderiv/root_termination.html">Termination
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Condition Functors</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/roots_noderiv/implementation.html">Implementation</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/roots_deriv.html">Root Finding With Derivatives:
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Newton-Raphson, Halley & Schröder</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/cubic_roots.html">Roots of Cubic Polynomials</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/quartic_roots.html">Roots of Quartic Polynomials</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/root_finding_examples.html">Examples of Root-Finding
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(with and without derivatives)</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/root_finding_examples/cbrt_eg.html">Finding the
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Cubed Root With and Without Derivatives</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/root_finding_examples/lambda.html">Using C++11
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Lambda's</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/root_finding_examples/5th_root_eg.html">Computing
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the Fifth Root</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/root_finding_examples/multiprecision_root.html">Root-finding
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using Boost.Multiprecision</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/root_finding_examples/nth_root.html">Generalizing
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to Compute the nth root</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/root_finding_examples/elliptic_eg.html">A More
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complex example - Inverting the Elliptic Integrals</a></span></dt>
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</dl></dd>
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<dt><span class="section"><a href="math_toolkit/bad_guess.html">The Effect of a Poor Initial Guess</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/bad_roots.html">Examples Where Root Finding Goes
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Wrong</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/brent_minima.html">Locating Function Minima using
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Brent's algorithm</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/root_comparison.html">Comparison of Root Finding
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Algorithms</a></span></dt>
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<dd><dl>
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<dt><span class="section"><a href="math_toolkit/root_comparison/cbrt_comparison.html">Comparison
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of Cube Root Finding Algorithms</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/root_comparison/root_n_comparison.html">Comparison
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of Nth-root Finding Algorithms</a></span></dt>
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<dt><span class="section"><a href="math_toolkit/root_comparison/elliptic_comparison.html">Comparison
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of Elliptic Integral Root Finding Algorithms</a></span></dt>
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</dl></dd>
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</dl>
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</div>
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<p>
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Several tools are provided to aid finding minima and roots of functions.
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</p>
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<p>
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Some <a class="link" href="math_toolkit/roots_noderiv.html" title="Root Finding Without Derivatives">root-finding without derivatives</a>
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methods are <a class="link" href="math_toolkit/roots_noderiv/bisect.html" title="Bisection">bisection</a>,
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<a class="link" href="math_toolkit/roots_noderiv/bracket_solve.html" title="Bracket and Solve Root">bracket and solve</a>,
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including use of <a class="link" href="math_toolkit/roots_noderiv/TOMS748.html" title="Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions">TOMS 748
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algorithm</a>.
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</p>
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<p>
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For <a class="link" href="math_toolkit/roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley & Schröder">root-finding with derivatives</a>
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the methods of <a class="link" href="math_toolkit/roots_deriv.html#math_toolkit.roots_deriv.newton">Newton-Raphson
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iteration</a>, <a class="link" href="math_toolkit/roots_deriv.html#math_toolkit.roots_deriv.halley">Halley</a>,
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and <a class="link" href="math_toolkit/roots_deriv.html#math_toolkit.roots_deriv.schroder">Schröder</a> are implemented.
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</p>
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<p>
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For locating minima of a function, a <a class="link" href="math_toolkit/brent_minima.html#math_toolkit.brent_minima.example">Brent
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minima finding example</a> is provided.
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</p>
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<p>
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There are several fully-worked <a class="link" href="math_toolkit/root_finding_examples.html" title="Examples of Root-Finding (with and without derivatives)">root-finding
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examples</a>, including:
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</p>
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<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
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<li class="listitem">
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<a class="link" href="math_toolkit/root_finding_examples/cbrt_eg.html#math_toolkit.root_finding_examples.cbrt_eg.cbrt_no_derivatives">root-finding
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without derivatives</a>
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</li>
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<li class="listitem">
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<a class="link" href="math_toolkit/root_finding_examples/cbrt_eg.html#math_toolkit.root_finding_examples.cbrt_eg.cbrt_1st_derivative">root-finding
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with 1st derivatives</a>
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</li>
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<li class="listitem">
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<a class="link" href="math_toolkit/root_finding_examples/cbrt_eg.html#math_toolkit.root_finding_examples.cbrt_eg.cbrt_2_derivatives">root-finding
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with 1st and 2nd derivatives</a>
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</li>
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</ul></div>
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</div>
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<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
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Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
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Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
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Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
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Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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