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