1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
|
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
<title>Locating Function Minima: Brent's algorithm</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.78.1">
<link rel="home" href="../../index.html" title="Math Toolkit 2.1.0">
<link rel="up" href="../internals1.html" title="Utilities & internals">
<link rel="prev" href="roots2.html" title="Root Finding Without Derivatives: Bisection, Bracket and TOMS748">
<link rel="next" href="tuples.html" title="Tuples">
</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="roots2.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals1.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="tuples.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.internals1.minima"></a><a class="link" href="minima.html" title="Locating Function Minima: Brent's algorithm">Locating Function Minima:
Brent's algorithm</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.internals1.minima.h0"></a>
<span class="phrase"><a name="math_toolkit.internals1.minima.synopsis"></a></span><a class="link" href="minima.html#math_toolkit.internals1.minima.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">minima</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
</pre>
<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="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">brent_find_minima</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="keyword">int</span> <span class="identifier">bits</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="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">brent_find_minima</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="keyword">int</span> <span class="identifier">bits</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>
</pre>
<h5>
<a name="math_toolkit.internals1.minima.h1"></a>
<span class="phrase"><a name="math_toolkit.internals1.minima.description"></a></span><a class="link" href="minima.html#math_toolkit.internals1.minima.description">Description</a>
</h5>
<p>
These two functions locate the minima of the continuous function <span class="emphasis"><em>f</em></span>
using Brent's algorithm. Parameters are:
</p>
<div class="variablelist">
<p class="title"><b></b></p>
<dl class="variablelist">
<dt><span class="term">f</span></dt>
<dd><p>
The function to minimise. The function should be smooth over the range
[min,max], with no maxima occurring in that interval.
</p></dd>
<dt><span class="term">min</span></dt>
<dd><p>
The lower endpoint of the range in which to search for the minima.
</p></dd>
<dt><span class="term">max</span></dt>
<dd><p>
The upper endpoint of the range in which to search for the minima.
</p></dd>
<dt><span class="term">bits</span></dt>
<dd><p>
The number of bits precision to which the minima should be found. Note
that in principle, the minima can not be located to greater accuracy
than the square root of machine epsilon (for 64-bit double, sqrt(1e-16)≅1e-8),
therefore if <span class="emphasis"><em>bits</em></span> is set to a value greater than
one half of the bits in type T, then the value will be ignored.
</p></dd>
<dt><span class="term">max_iter</span></dt>
<dd><p>
The maximum number of iterations to use in the algorithm, if not provided
the algorithm will just keep on going until the minima is found.
</p></dd>
</dl>
</div>
<p>
<span class="bold"><strong>Returns:</strong></span> a pair containing the value of
the abscissa at the minima and the value of f(x) at the minima.
</p>
<h5>
<a name="math_toolkit.internals1.minima.h2"></a>
<span class="phrase"><a name="math_toolkit.internals1.minima.implementation"></a></span><a class="link" href="minima.html#math_toolkit.internals1.minima.implementation">Implementation</a>
</h5>
<p>
This is a reasonably faithful implementation of Brent's algorithm, refer
to:
</p>
<p>
Brent, R.P. 1973, Algorithms for Minimization without Derivatives (Englewood
Cliffs, NJ: Prentice-Hall), Chapter 5.
</p>
<p>
Numerical Recipes in C, The Art of Scientific Computing, Second Edition,
William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P.
Flannery. Cambridge University Press. 1988, 1992.
</p>
<p>
An algorithm with guaranteed convergence for finding a zero of a function,
R. P. Brent, The Computer Journal, Vol 44, 1971.
</p>
</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 © 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta,
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="roots2.html"><img src="../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../internals1.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="tuples.html"><img src="../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>
|
