123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219220 |
- <html>
- <head>
- <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
- <title>Root Finding Without Derivatives</title>
- <link rel="stylesheet" href="../math.css" type="text/css">
- <meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
- <link rel="home" href="../index.html" title="Math Toolkit 2.11.0">
- <link rel="up" href="../root_finding.html" title="Chapter 10. Root Finding & Minimization Algorithms">
- <link rel="prev" href="../root_finding.html" title="Chapter 10. Root Finding & Minimization Algorithms">
- <link rel="next" href="roots_noderiv/bisect.html" title="Bisection">
- </head>
- <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
- <table cellpadding="2" width="100%"><tr>
- <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td>
- <td align="center"><a href="../../../../../index.html">Home</a></td>
- <td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td>
- <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
- <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
- <td align="center"><a href="../../../../../more/index.htm">More</a></td>
- </tr></table>
- <hr>
- <div class="spirit-nav">
- <a accesskey="p" href="../root_finding.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../root_finding.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="roots_noderiv/bisect.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
- </div>
- <div class="section">
- <div class="titlepage"><div><div><h2 class="title" style="clear: both">
- <a name="math_toolkit.roots_noderiv"></a><a class="link" href="roots_noderiv.html" title="Root Finding Without Derivatives">Root Finding Without Derivatives</a>
- </h2></div></div></div>
- <div class="toc"><dl class="toc">
- <dt><span class="section"><a href="roots_noderiv/bisect.html">Bisection</a></span></dt>
- <dt><span class="section"><a href="roots_noderiv/bracket_solve.html">Bracket and
- Solve Root</a></span></dt>
- <dt><span class="section"><a href="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="roots_noderiv/brent.html">Brent-Decker Algorithm</a></span></dt>
- <dt><span class="section"><a href="roots_noderiv/root_termination.html">Termination
- Condition Functors</a></span></dt>
- <dt><span class="section"><a href="roots_noderiv/implementation.html">Implementation</a></span></dt>
- </dl></div>
- <h5>
- <a name="math_toolkit.roots_noderiv.h0"></a>
- <span class="phrase"><a name="math_toolkit.roots_noderiv.synopsis"></a></span><a class="link" href="roots_noderiv.html#math_toolkit.roots_noderiv.synopsis">Synopsis</a>
- </h5>
- <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">roots</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
- </pre>
- <pre class="programlisting"> <span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span>
- <span class="keyword">namespace</span> <span class="identifier">tools</span> <span class="special">{</span> <span class="comment">// Note namespace boost::math::tools.</span>
- <span class="comment">// Bisection</span>
- <span class="keyword">template</span> <span class="special"><</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="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
- <span class="identifier">bisect</span><span class="special">(</span>
- <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
- <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
- <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
- <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
- <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span>
- <span class="keyword">template</span> <span class="special"><</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="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
- <span class="identifier">bisect</span><span class="special">(</span>
- <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
- <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
- <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
- <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">);</span>
- <span class="keyword">template</span> <span class="special"><</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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
- <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
- <span class="identifier">bisect</span><span class="special">(</span>
- <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
- <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span>
- <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span>
- <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
- <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">,</span>
- <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
- <span class="comment">// Bracket and Solve Root</span>
- <span class="keyword">template</span> <span class="special"><</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="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
- <span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
- <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
- <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">guess</span><span class="special">,</span>
- <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span>
- <span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
- <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
- <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span>
- <span class="keyword">template</span> <span class="special"><</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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
- <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
- <span class="identifier">bracket_and_solve_root</span><span class="special">(</span>
- <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
- <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">guess</span><span class="special">,</span>
- <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">factor</span><span class="special">,</span>
- <span class="keyword">bool</span> <span class="identifier">rising</span><span class="special">,</span>
- <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
- <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">,</span>
- <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
- <span class="comment">// TOMS 748 algorithm</span>
- <span class="keyword">template</span> <span class="special"><</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="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
- <span class="identifier">toms748_solve</span><span class="special">(</span>
- <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
- <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
- <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
- <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
- <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span>
- <span class="keyword">template</span> <span class="special"><</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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
- <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
- <span class="identifier">toms748_solve</span><span class="special">(</span>
- <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
- <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
- <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
- <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
- <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">,</span>
- <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
- <span class="keyword">template</span> <span class="special"><</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="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
- <span class="identifier">toms748_solve</span><span class="special">(</span>
- <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
- <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
- <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
- <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span>
- <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span>
- <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
- <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span>
- <span class="keyword">template</span> <span class="special"><</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 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span>
- <span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span>
- <span class="identifier">toms748_solve</span><span class="special">(</span>
- <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
- <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span>
- <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span>
- <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span>
- <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span>
- <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span>
- <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">,</span>
- <span class="keyword">const</span> <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span>
- <span class="comment">// Termination conditions:</span>
- <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
- <span class="keyword">struct</span> <span class="identifier">eps_tolerance</span><span class="special">;</span>
- <span class="keyword">struct</span> <span class="identifier">equal_floor</span><span class="special">;</span>
- <span class="keyword">struct</span> <span class="identifier">equal_ceil</span><span class="special">;</span>
- <span class="keyword">struct</span> <span class="identifier">equal_nearest_integer</span><span class="special">;</span>
- <span class="special">}}}</span> <span class="comment">// boost::math::tools namespaces</span>
- </pre>
- <h5>
- <a name="math_toolkit.roots_noderiv.h1"></a>
- <span class="phrase"><a name="math_toolkit.roots_noderiv.description"></a></span><a class="link" href="roots_noderiv.html#math_toolkit.roots_noderiv.description">Description</a>
- </h5>
- <p>
- These functions solve the root of some function <span class="emphasis"><em>f(x)</em></span> -
- <span class="emphasis"><em>without the need for any derivatives of <span class="emphasis"><em>f(x)</em></span></em></span>.
- </p>
- <p>
- The <code class="computeroutput"><span class="identifier">bracket_and_solve_root</span></code>
- functions use <a class="link" href="roots_noderiv/TOMS748.html" title="Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions">TOMS 748 algorithm</a>
- by Alefeld, Potra and Shi that is asymptotically the most efficient known,
- and has been shown to be optimal for a certain classes of smooth functions.
- Variants with and without <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policies</a> are provided.
- </p>
- <p>
- Alternatively, <a class="link" href="roots_noderiv/bisect.html" title="Bisection">bisect</a>
- is a simple <a href="https://en.wikipedia.org/wiki/Bisection" target="_top">bisection</a>
- routine which can be useful in its own right in some situations, or alternatively
- for narrowing down the range containing the root, prior to calling a more advanced
- algorithm.
- </p>
- <p>
- All the algorithms in this section reduce the diameter of the enclosing interval
- with the same asymptotic efficiency with which they locate the root. This is
- in contrast to the derivative based methods which may <span class="emphasis"><em>never</em></span>
- significantly reduce the enclosing interval, even though they rapidly approach
- the root. This is also in contrast to some other derivative-free methods (for
- example, Brent's method described at <a href="http://en.wikipedia.org/wiki/Brent%27s_method" target="_top">Brent-Dekker)</a>
- which only reduces the enclosing interval on the final step. Therefore these
- methods return a <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span></code> containing the enclosing interval found,
- and accept a function object specifying the termination condition.
- </p>
- <p>
- Three function objects are provided for ready-made termination conditions:
- </p>
- <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
- <li class="listitem">
- <span class="emphasis"><em>eps_tolerance</em></span> causes termination when the relative
- error in the enclosing interval is below a certain threshold.
- </li>
- <li class="listitem">
- <span class="emphasis"><em>equal_floor</em></span> and <span class="emphasis"><em>equal_ceil</em></span> are
- useful for certain statistical applications where the result is known to
- be an integer.
- </li>
- <li class="listitem">
- Other user-defined termination conditions are likely to be used only rarely,
- but may be useful in some specific circumstances.
- </li>
- </ul></div>
- </div>
- <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
- <td align="left"></td>
- <td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
- Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
- Hubert Holin, Bruno Lalande, John Maddock, 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
- 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>)
- </p>
- </div></td>
- </tr></table>
- <hr>
- <div class="spirit-nav">
- <a accesskey="p" href="../root_finding.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../root_finding.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="roots_noderiv/bisect.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
- </div>
- </body>
- </html>
|