1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions</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 2.12.0"> 8<link rel="up" href="../roots_noderiv.html" title="Root Finding Without Derivatives"> 9<link rel="prev" href="bracket_solve.html" title="Bracket and Solve Root"> 10<link rel="next" href="brent.html" title="Brent-Decker Algorithm"> 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="bracket_solve.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="brent.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.TOMS748"></a><a class="link" href="TOMS748.html" title="Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions">Algorithm TOMS 748: 28 Alefeld, Potra and Shi: Enclosing zeros of continuous functions</a> 29</h3></div></div></div> 30<pre class="programlisting"><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> 31<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> 32 <span class="identifier">toms748_solve</span><span class="special">(</span> 33 <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> 34 <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> 35 <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> 36 <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> 37 <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> 38 39<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 40<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> 41 <span class="identifier">toms748_solve</span><span class="special">(</span> 42 <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> 43 <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> 44 <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> 45 <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> 46 <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> 47 <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 48 49<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> 50<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> 51 <span class="identifier">toms748_solve</span><span class="special">(</span> 52 <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> 53 <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> 54 <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> 55 <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span> 56 <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span> 57 <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> 58 <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> 59 60<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 61<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> 62 <span class="identifier">toms748_solve</span><span class="special">(</span> 63 <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> 64 <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">a</span><span class="special">,</span> 65 <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">b</span><span class="special">,</span> 66 <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fa</span><span class="special">,</span> 67 <span class="keyword">const</span> <span class="identifier">T</span><span class="special">&</span> <span class="identifier">fb</span><span class="special">,</span> 68 <span class="identifier">Tol</span> <span class="identifier">tol</span><span class="special">,</span> 69 <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> 70 <span class="keyword">const</span> <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&);</span> 71</pre> 72<p> 73 These functions implement TOMS Algorithm 748: it uses a mixture of cubic, 74 quadratic and linear (secant) interpolation to locate the root of <span class="emphasis"><em>f(x)</em></span>. 75 The two pairs of functions differ only by whether values for <span class="emphasis"><em>f(a)</em></span> 76 and <span class="emphasis"><em>f(b)</em></span> are already available. 77 </p> 78<p> 79 Generally speaking it is easier (and often more efficient) to use <a class="link" href="bracket_solve.html" title="Bracket and Solve Root">bracket 80 and solve</a> rather than trying to bracket the root yourself as this 81 function requires. 82 </p> 83<p> 84 This function is provided rather than <a href="http://en.wikipedia.org/wiki/Brent%27s_method" target="_top">Brent's 85 method</a> as it is known to be more efficient in many cases (it is asymptotically 86 the most efficient known, and has been shown to be optimal for a certain 87 classes of smooth functions). It also has the useful property of decreasing 88 the bracket size with each step, unlike Brent's method which only shrinks 89 the enclosing interval in the final step. This makes it particularly useful 90 when you need a result where the ends of the interval round to the same integer: 91 as often happens in statistical applications, for example. In this situation 92 the function is able to exit after a much smaller number of iterations than 93 would otherwise be possible. 94 </p> 95<p> 96 The <a class="link" href="TOMS748.html" title="Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions">TOMS 748 algorithm</a> 97 parameters are: 98 </p> 99<div class="variablelist"> 100<p class="title"><b></b></p> 101<dl class="variablelist"> 102<dt><span class="term">f</span></dt> 103<dd><p> 104 A unary functor (or C++ lambda) that is the function whose root is 105 to be solved. f(x) need not be uniformly increasing or decreasing on 106 <span class="emphasis"><em>x</em></span> and may have multiple roots. However, the bounds 107 given must bracket a single root. 108 </p></dd> 109<dt><span class="term">a</span></dt> 110<dd><p> 111 The lower bound for the initial bracket of the root. 112 </p></dd> 113<dt><span class="term">b</span></dt> 114<dd><p> 115 The upper bound for the initial bracket of the root. It is a precondition 116 that <span class="emphasis"><em>a < b</em></span> and that <span class="emphasis"><em>a</em></span> 117 and <span class="emphasis"><em>b</em></span> bracket the root to find so that <span class="emphasis"><em>f(a) 118 * f(b) < 0</em></span>. 119 </p></dd> 120<dt><span class="term">fa</span></dt> 121<dd><p> 122 Optional: the value of <span class="emphasis"><em>f(a)</em></span>. 123 </p></dd> 124<dt><span class="term">fb</span></dt> 125<dd><p> 126 Optional: the value of <span class="emphasis"><em>f(b)</em></span>. 127 </p></dd> 128<dt><span class="term">tol</span></dt> 129<dd><p> 130 A binary functor (or C++ lambda) that determines the termination condition 131 for the search for the root. <span class="emphasis"><em>tol</em></span> is passed the 132 current brackets at each step, when it returns true, then the current 133 brackets are returned as the result. See also <a class="link" href="root_termination.html" title="Termination Condition Functors">predefined 134 termination functors</a>. 135 </p></dd> 136<dt><span class="term">max_iter</span></dt> 137<dd><p> 138 The maximum number of function invocations to perform in the search 139 for the root. On exit, <span class="emphasis"><em>max_iter</em></span> is set to actual 140 number of function invocations used. 141 </p></dd> 142</dl> 143</div> 144<p> 145 The final <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and can 146 be used to control the behaviour of the function: how it handles errors, 147 what level of precision to use etc. Refer to the <a class="link" href="../../policy.html" title="Chapter 21. Policies: Controlling Precision, Error Handling etc">policy 148 documentation for more details</a>. 149 </p> 150<p> 151 <code class="computeroutput"><span class="identifier">toms748_solve</span></code> returns: a 152 pair of values <span class="emphasis"><em>r</em></span> that bracket the root so that: 153 </p> 154<div class="blockquote"><blockquote class="blockquote"><p> 155 <span class="emphasis"><em>f(r.first) * f(r.second) <= 0</em></span> 156 </p></blockquote></div> 157<p> 158 and either 159 </p> 160<div class="blockquote"><blockquote class="blockquote"><p> 161 <span class="emphasis"><em>tol(r.first, r.second) == true</em></span> 162 </p></blockquote></div> 163<p> 164 or 165 </p> 166<div class="blockquote"><blockquote class="blockquote"><p> 167 <span class="emphasis"><em>max_iter >= m</em></span> 168 </p></blockquote></div> 169<p> 170 where <span class="emphasis"><em>m</em></span> is the initial value of <span class="emphasis"><em>max_iter</em></span> 171 passed to the function. 172 </p> 173<p> 174 In other words, it's up to the caller to verify whether termination occurred 175 as a result of exceeding <span class="emphasis"><em>max_iter</em></span> function invocations 176 (easily done by checking the updated value of <span class="emphasis"><em>max_iter</em></span> 177 against its previous value passed as parameter), rather than because the 178 termination condition <span class="emphasis"><em>tol</em></span> was satisfied. 179 </p> 180</div> 181<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 182<td align="left"></td> 183<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 184 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 185 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 186 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 187 Daryle Walker and Xiaogang Zhang<p> 188 Distributed under the Boost Software License, Version 1.0. (See accompanying 189 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>) 190 </p> 191</div></td> 192</tr></table> 193<hr> 194<div class="spirit-nav"> 195<a accesskey="p" href="bracket_solve.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="brent.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a> 196</div> 197</body> 198</html> 199
