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  26 <div class="titlepage"><div><div><h1 class="title">
  27 <a name="root_finding"></a>Chapter&#160;10.&#160;Root Finding &amp; Minimization Algorithms</h1></div></div></div>
  28 <div class="toc">
  29 <p><b>Table of Contents</b></p>
  30 <dl class="toc">
  31 <dt><span class="section"><a href="math_toolkit/roots_noderiv.html">Root Finding Without Derivatives</a></span></dt>
  32 <dd><dl>
  33 <dt><span class="section"><a href="math_toolkit/roots_noderiv/bisect.html">Bisection</a></span></dt>
  34 <dt><span class="section"><a href="math_toolkit/roots_noderiv/bracket_solve.html">Bracket and
  35       Solve Root</a></span></dt>
  36 <dt><span class="section"><a href="math_toolkit/roots_noderiv/TOMS748.html">Algorithm TOMS 748:
  37       Alefeld, Potra and Shi: Enclosing zeros of continuous functions</a></span></dt>
  38 <dt><span class="section"><a href="math_toolkit/roots_noderiv/brent.html">Brent-Decker Algorithm</a></span></dt>
  39 <dt><span class="section"><a href="math_toolkit/roots_noderiv/root_termination.html">Termination
  40       Condition Functors</a></span></dt>
  41 <dt><span class="section"><a href="math_toolkit/roots_noderiv/implementation.html">Implementation</a></span></dt>
  42 </dl></dd>
  43 <dt><span class="section"><a href="math_toolkit/roots_deriv.html">Root Finding With Derivatives:
  44     Newton-Raphson, Halley &amp; Schr&#246;der</a></span></dt>
  45 <dt><span class="section"><a href="math_toolkit/root_finding_examples.html">Examples of Root-Finding
  46     (with and without derivatives)</a></span></dt>
  47 <dd><dl>
  48 <dt><span class="section"><a href="math_toolkit/root_finding_examples/cbrt_eg.html">Finding the
  49       Cubed Root With and Without Derivatives</a></span></dt>
  50 <dt><span class="section"><a href="math_toolkit/root_finding_examples/lambda.html">Using C++11
  51       Lambda's</a></span></dt>
  52 <dt><span class="section"><a href="math_toolkit/root_finding_examples/5th_root_eg.html">Computing
  53       the Fifth Root</a></span></dt>
  54 <dt><span class="section"><a href="math_toolkit/root_finding_examples/multiprecision_root.html">Root-finding
  55       using Boost.Multiprecision</a></span></dt>
  56 <dt><span class="section"><a href="math_toolkit/root_finding_examples/nth_root.html">Generalizing
  57       to Compute the nth root</a></span></dt>
  58 <dt><span class="section"><a href="math_toolkit/root_finding_examples/elliptic_eg.html">A More
  59       complex example - Inverting the Elliptic Integrals</a></span></dt>
  60 </dl></dd>
  61 <dt><span class="section"><a href="math_toolkit/bad_guess.html">The Effect of a Poor Initial Guess</a></span></dt>
  62 <dt><span class="section"><a href="math_toolkit/bad_roots.html">Examples Where Root Finding Goes
  63     Wrong</a></span></dt>
  64 <dt><span class="section"><a href="math_toolkit/brent_minima.html">Locating Function Minima using
  65     Brent's algorithm</a></span></dt>
  66 <dt><span class="section"><a href="math_toolkit/root_comparison.html">Comparison of Root Finding
  67     Algorithms</a></span></dt>
  68 <dd><dl>
  69 <dt><span class="section"><a href="math_toolkit/root_comparison/cbrt_comparison.html">Comparison
  70       of Cube Root Finding Algorithms</a></span></dt>
  71 <dt><span class="section"><a href="math_toolkit/root_comparison/root_n_comparison.html">Comparison
  72       of Nth-root Finding Algorithms</a></span></dt>
  73 <dt><span class="section"><a href="math_toolkit/root_comparison/elliptic_comparison.html">Comparison
  74       of Elliptic Integral Root Finding Algoritghms</a></span></dt>
  75 </dl></dd>
  76 </dl>
  77 </div>
  78 <p>
  79     Several tools are provided to aid finding minima and roots of functions.
  80   </p>
  81 <p>
  82     Some <a class="link" href="math_toolkit/roots_noderiv.html" title="Root Finding Without Derivatives">root-finding without derivatives</a>
  83     methods are <a class="link" href="math_toolkit/roots_noderiv/bisect.html" title="Bisection">bisection</a>,
  84     <a class="link" href="math_toolkit/roots_noderiv/bracket_solve.html" title="Bracket and Solve Root">bracket and solve</a>,
  85     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
  86     algorithm</a>.
  87   </p>
  88 <p>
  89     For <a class="link" href="math_toolkit/roots_deriv.html" title="Root Finding With Derivatives: Newton-Raphson, Halley &amp; Schr&#246;der">root-finding with derivatives</a>
  90     the methods of <a class="link" href="math_toolkit/roots_deriv.html#math_toolkit.roots_deriv.newton">Newton-Raphson
  91     iteration</a>, <a class="link" href="math_toolkit/roots_deriv.html#math_toolkit.roots_deriv.halley">Halley</a>,
  92     and <a class="link" href="math_toolkit/roots_deriv.html#math_toolkit.roots_deriv.schroder">Schr&#246;der</a> are implemented.
  93   </p>
  94 <p>
  95     For locating minima of a function, a <a class="link" href="math_toolkit/brent_minima.html#math_toolkit.brent_minima.example">Brent
  96     minima finding example</a> is provided.
  97   </p>
  98 <p>
  99     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
 100     examples</a>, including:
 101   </p>
 102 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
 103 <li class="listitem">
 104         <a class="link" href="math_toolkit/root_finding_examples/cbrt_eg.html#math_toolkit.root_finding_examples.cbrt_eg.cbrt_no_derivatives">root-finding
 105         without derivatives</a>
 106       </li>
 107 <li class="listitem">
 108         <a class="link" href="math_toolkit/root_finding_examples/cbrt_eg.html#math_toolkit.root_finding_examples.cbrt_eg.cbrt_1st_derivative">root-finding
 109         with 1st derivatives</a>
 110       </li>
 111 <li class="listitem">
 112         <a class="link" href="math_toolkit/root_finding_examples/cbrt_eg.html#math_toolkit.root_finding_examples.cbrt_eg.cbrt_2_derivatives">root-finding
 113         with 1st and 2nd derivatives</a>
 114       </li>
 115 </ul></div>
 116 </div>
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