]>
Commit | Line | Data |
---|---|---|
7c673cae FG |
1 | <html> |
2 | <head> | |
3 | <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> | |
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 V1.77.1"> | |
7 | <link rel="home" href="../../../index.html" title="Math Toolkit 2.5.1"> | |
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><h4 class="title"> | |
27 | <a name="math_toolkit.roots.roots_noderiv.TOMS748"></a><a class="link" href="TOMS748.html" title="Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions">Algorithm | |
28 | TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions</a> | |
29 | </h4></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 15. 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 15. 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 15. 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 15. 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 and solve</a> | |
80 | rather than trying to bracket the root yourself as this function requires. | |
81 | </p> | |
82 | <p> | |
83 | This function is provided rather than <a href="http://en.wikipedia.org/wiki/Brent%27s_method" target="_top">Brent's | |
84 | method</a> as it is known to be more effient in many cases (it is asymptotically | |
85 | the most efficient known, and has been shown to be optimal for a certain | |
86 | classes of smooth functions). It also has the useful property of decreasing | |
87 | the bracket size with each step, unlike Brent's method which only shrinks | |
88 | the enclosing interval in the final step. This makes it particularly useful | |
89 | when you need a result where the ends of the interval round to the same | |
90 | integer: as often happens in statistical applications for example. In this | |
91 | situation the function is able to exit after a much smaller number of iterations | |
92 | than would otherwise be possible. | |
93 | </p> | |
94 | <p> | |
95 | The <a class="link" href="TOMS748.html" title="Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions">TOMS 748 algorithm</a> | |
96 | parameters are: | |
97 | </p> | |
98 | <div class="variablelist"> | |
99 | <p class="title"><b></b></p> | |
100 | <dl class="variablelist"> | |
101 | <dt><span class="term">f</span></dt> | |
102 | <dd><p> | |
103 | A unary functor that is the function whose root is to be solved. | |
104 | f(x) need not be uniformly increasing or decreasing on <span class="emphasis"><em>x</em></span> | |
105 | and may have multiple roots. However, the bounds given must bracket | |
106 | a single root. | |
107 | </p></dd> | |
108 | <dt><span class="term">a</span></dt> | |
109 | <dd><p> | |
110 | The lower bound for the initial bracket of the root. | |
111 | </p></dd> | |
112 | <dt><span class="term">b</span></dt> | |
113 | <dd><p> | |
114 | The upper bound for the initial bracket of the root. It is a precondition | |
115 | that <span class="emphasis"><em>a < b</em></span> and that <span class="emphasis"><em>a</em></span> | |
116 | and <span class="emphasis"><em>b</em></span> bracket the root to find so that <span class="emphasis"><em>f(a) | |
117 | * f(b) < 0</em></span>. | |
118 | </p></dd> | |
119 | <dt><span class="term">fa</span></dt> | |
120 | <dd><p> | |
121 | Optional: the value of <span class="emphasis"><em>f(a)</em></span>. | |
122 | </p></dd> | |
123 | <dt><span class="term">fb</span></dt> | |
124 | <dd><p> | |
125 | Optional: the value of <span class="emphasis"><em>f(b)</em></span>. | |
126 | </p></dd> | |
127 | <dt><span class="term">tol</span></dt> | |
128 | <dd><p> | |
129 | A binary functor that determines the termination condition for the | |
130 | search for the root. <span class="emphasis"><em>tol</em></span> is passed the current | |
131 | brackets at each step, when it returns true, then the current brackets | |
132 | are returned as the result. See also <a class="link" href="root_termination.html" title="Termination Condition Functors">predefined | |
133 | termination functors</a>. | |
134 | </p></dd> | |
135 | <dt><span class="term">max_iter</span></dt> | |
136 | <dd><p> | |
137 | The maximum number of function invocations to perform in the search | |
138 | for the root. On exit, <span class="emphasis"><em>max_iter</em></span> is set to actual | |
139 | number of function invocations used. | |
140 | </p></dd> | |
141 | </dl> | |
142 | </div> | |
143 | <p> | |
144 | The final <a class="link" href="../../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">Policy</a> argument is optional and | |
145 | can be used to control the behaviour of the function: how it handles errors, | |
146 | what level of precision to use etc. Refer to the <a class="link" href="../../../policy.html" title="Chapter 15. Policies: Controlling Precision, Error Handling etc">policy | |
147 | documentation for more details</a>. | |
148 | </p> | |
149 | <p> | |
150 | <code class="computeroutput"><span class="identifier">toms748_solve</span></code> returns: | |
151 | a pair of values <span class="emphasis"><em>r</em></span> that bracket the root so that: | |
152 | </p> | |
153 | <pre class="programlisting"><span class="identifier">f</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">f</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span><span class="special">)</span> <span class="special"><=</span> <span class="number">0</span> | |
154 | </pre> | |
155 | <p> | |
156 | and either | |
157 | </p> | |
158 | <pre class="programlisting"><span class="identifier">tol</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">,</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span><span class="special">)</span> <span class="special">==</span> <span class="keyword">true</span> | |
159 | </pre> | |
160 | <p> | |
161 | or | |
162 | </p> | |
163 | <pre class="programlisting"><span class="identifier">max_iter</span> <span class="special">>=</span> <span class="identifier">m</span> | |
164 | </pre> | |
165 | <p> | |
166 | where <span class="emphasis"><em>m</em></span> is the initial value of <span class="emphasis"><em>max_iter</em></span> | |
167 | passed to the function. | |
168 | </p> | |
169 | <p> | |
170 | In other words, it's up to the caller to verify whether termination occurred | |
171 | as a result of exceeding <span class="emphasis"><em>max_iter</em></span> function invocations | |
172 | (easily done by checking the updated value of <span class="emphasis"><em>max_iter</em></span> | |
173 | against its previous value passed as parameter), rather than because the | |
174 | termination condition <span class="emphasis"><em>tol</em></span> was satisfied. | |
175 | </p> | |
176 | </div> | |
177 | <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> | |
178 | <td align="left"></td> | |
179 | <td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014 Nikhar Agrawal, | |
180 | Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert | |
181 | Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan Råde, Gautam Sewani, | |
182 | Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p> | |
183 | Distributed under the Boost Software License, Version 1.0. (See accompanying | |
184 | 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>) | |
185 | </p> | |
186 | </div></td> | |
187 | </tr></table> | |
188 | <hr> | |
189 | <div class="spirit-nav"> | |
190 | <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> | |
191 | </div> | |
192 | </body> | |
193 | </html> |