libs/math/doc/html/math_toolkit/roots_noderiv/bisect.html

   1 <html>
   2 <head>
   3 <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
   4 <title>Bisection</title>
   5 <link rel="stylesheet" href="../../math.css" type="text/css">
   6 <meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
   7 <link rel="home" href="../../index.html" title="Math Toolkit 2.11.0">
   8 <link rel="up" href="../roots_noderiv.html" title="Root Finding Without Derivatives">
   9 <link rel="prev" href="../roots_noderiv.html" title="Root Finding Without Derivatives">
  10 <link rel="next" href="bracket_solve.html" title="Bracket and Solve Root">
  11 </head>
  12 <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
  13 <table cellpadding="2" width="100%"><tr>
  14 <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../boost.png"></td>
  15 <td align="center"><a href="../../../../../../index.html">Home</a></td>
  16 <td align="center"><a href="../../../../../../libs/libraries.htm">Libraries</a></td>
  17 <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
  18 <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
  19 <td align="center"><a href="../../../../../../more/index.htm">More</a></td>
  20 </tr></table>
  21 <hr>
  22 <div class="spirit-nav">
  23 <a accesskey="p" href="../roots_noderiv.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../roots_noderiv.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="bracket_solve.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
  24 </div>
  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.roots_noderiv.bisect"></a><a class="link" href="bisect.html" title="Bisection">Bisection</a>
  28 </h3></div></div></div>
  29 <pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">&gt;</span>
  30 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</span>
  31    <span class="identifier">bisect</span><span class="special">(</span>  <span class="comment">// Unlimited iterations.</span>
  32       <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
  33       <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
  34       <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
  35       <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">);</span>
  36
  37 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">&gt;</span>
  38 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</span>
  39    <span class="identifier">bisect</span><span class="special">(</span>  <span class="comment">// Limited iterations.</span>
  40       <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
  41       <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
  42       <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
  43       <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
  44       <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_iter</span><span class="special">);</span>
  45
  46 <span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">Tol</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  47 <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">&gt;</span>
  48    <span class="identifier">bisect</span><span class="special">(</span> <span class="comment">// Specified policy.</span>
  49       <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
  50       <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
  51       <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
  52       <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
  53       <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_iter</span><span class="special">,</span>
  54       <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&amp;);</span>
  55 </pre>
  56 <p>
  57         These functions locate the root using <a href="https://en.wikipedia.org/wiki/Bisection" target="_top">bisection</a>.
  58       </p>
  59 <p>
  60         <code class="computeroutput"><span class="identifier">bisect</span></code> function arguments
  61         are:
  62       </p>
  63 <div class="variablelist">
  64 <p class="title"><b></b></p>
  65 <dl class="variablelist">
  66 <dt><span class="term">f</span></dt>
  67 <dd><p>
  68               A unary functor (or C++ lambda) which is the function <span class="emphasis"><em>f(x)</em></span>
  69               whose root is to be found.
  70             </p></dd>
  71 <dt><span class="term">min</span></dt>
  72 <dd><p>
  73               The left bracket of the interval known to contain the root.
  74             </p></dd>
  75 <dt><span class="term">max</span></dt>
  76 <dd><p>
  77               The right bracket of the interval known to contain the root.<br>
  78               It is a precondition that <span class="emphasis"><em>min &lt; max</em></span> and <span class="emphasis"><em>f(min)*f(max)
  79               &lt;= 0</em></span>, the function raises an <a class="link" href="../error_handling.html#math_toolkit.error_handling.evaluation_error">evaluation_error</a>
  80               if these preconditions are violated. The action taken on error is controlled
  81               by the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> template argument: the
  82               default behavior is to throw a <span class="emphasis"><em>boost::math::evaluation_error</em></span>.
  83               If the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> is changed to not throw
  84               then it returns <span class="emphasis"><em>std::pair&lt;T&gt;(min, min)</em></span>.
  85             </p></dd>
  86 <dt><span class="term">tol</span></dt>
  87 <dd><p>
  88               A binary functor (or C++ lambda) that specifies the termination condition:
  89               the function will return the current brackets enclosing the root when
  90               <span class="emphasis"><em>tol(min, max)</em></span> becomes true. See also <a class="link" href="root_termination.html" title="Termination Condition Functors">predefined
  91               termination functors</a>.
  92             </p></dd>
  93 <dt><span class="term">max_iter</span></dt>
  94 <dd><p>
  95               The maximum number of invocations of <span class="emphasis"><em>f(x)</em></span> to make
  96               while searching for the root. On exit, this is updated to the actual
  97               number of invocations performed.
  98             </p></dd>
  99 </dl>
 100 </div>
 101 <p>
 102         The final <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can
 103         be used to control the behaviour of the function: how it handles errors,
 104         what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">policy
 105         documentation for more details</a>.
 106       </p>
 107 <p>
 108         <span class="bold"><strong>Returns</strong></span>: a pair of values <span class="emphasis"><em>r</em></span>
 109         that bracket the root so that:
 110       </p>
 111 <div class="blockquote"><blockquote class="blockquote"><p>
 112           f(r.first) * f(r.second) &lt;= 0
 113         </p></blockquote></div>
 114 <p>
 115         and either
 116       </p>
 117 <div class="blockquote"><blockquote class="blockquote"><p>
 118           tol(r.first, r.second) == true
 119         </p></blockquote></div>
 120 <p>
 121         or
 122       </p>
 123 <div class="blockquote"><blockquote class="blockquote"><p>
 124           max_iter &gt;= m
 125         </p></blockquote></div>
 126 <p>
 127         where <span class="emphasis"><em>m</em></span> is the initial value of <span class="emphasis"><em>max_iter</em></span>
 128         passed to the function.
 129       </p>
 130 <p>
 131         In other words, it's up to the caller to verify whether termination occurred
 132         as a result of exceeding <span class="emphasis"><em>max_iter</em></span> function invocations
 133         (easily done by checking the updated value of <span class="emphasis"><em>max_iter</em></span>
 134         when the function returns), rather than because the termination condition
 135         <span class="emphasis"><em>tol</em></span> was satisfied.
 136       </p>
 137 </div>
 138 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 139 <td align="left"></td>
 140 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 141       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 142       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 143       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 144       Daryle Walker and Xiaogang Zhang<p>
 145         Distributed under the Boost Software License, Version 1.0. (See accompanying
 146         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>)
 147       </p>
 148 </div></td>
 149 </tr></table>
 150 <hr>
 151 <div class="spirit-nav">
 152 <a accesskey="p" href="../roots_noderiv.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../roots_noderiv.html"><img src="../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../index.html"><img src="../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="bracket_solve.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
 153 </div>
 154 </body>
 155 </html>