File: PLAIN-Monte-Carlo.html

package info (click to toggle)
gsl-ref-html 2.3-1
  • links: PTS
  • area: non-free
  • in suites: bullseye, buster, sid
  • size: 6,876 kB
  • ctags: 4,574
  • sloc: makefile: 35
file content (142 lines) | stat: -rw-r--r-- 7,263 bytes parent folder | download
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
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
<!-- Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016 The GSL Team.

Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.3 or
any later version published by the Free Software Foundation; with the
Invariant Sections being "GNU General Public License" and "Free Software
Needs Free Documentation", the Front-Cover text being "A GNU Manual",
and with the Back-Cover Text being (a) (see below). A copy of the
license is included in the section entitled "GNU Free Documentation
License".

(a) The Back-Cover Text is: "You have the freedom to copy and modify this
GNU Manual." -->
<!-- Created by GNU Texinfo 5.1, http://www.gnu.org/software/texinfo/ -->
<head>
<title>GNU Scientific Library &ndash; Reference Manual: PLAIN Monte Carlo</title>

<meta name="description" content="GNU Scientific Library &ndash; Reference Manual: PLAIN Monte Carlo">
<meta name="keywords" content="GNU Scientific Library &ndash; Reference Manual: PLAIN Monte Carlo">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="makeinfo">
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<link href="index.html#Top" rel="start" title="Top">
<link href="Function-Index.html#Function-Index" rel="index" title="Function Index">
<link href="Monte-Carlo-Integration.html#Monte-Carlo-Integration" rel="up" title="Monte Carlo Integration">
<link href="MISER.html#MISER" rel="next" title="MISER">
<link href="Monte-Carlo-Interface.html#Monte-Carlo-Interface" rel="previous" title="Monte Carlo Interface">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
blockquote.smallquotation {font-size: smaller}
div.display {margin-left: 3.2em}
div.example {margin-left: 3.2em}
div.indentedblock {margin-left: 3.2em}
div.lisp {margin-left: 3.2em}
div.smalldisplay {margin-left: 3.2em}
div.smallexample {margin-left: 3.2em}
div.smallindentedblock {margin-left: 3.2em; font-size: smaller}
div.smalllisp {margin-left: 3.2em}
kbd {font-style:oblique}
pre.display {font-family: inherit}
pre.format {font-family: inherit}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: inherit; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: inherit; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.nocodebreak {white-space:nowrap}
span.nolinebreak {white-space:nowrap}
span.roman {font-family:serif; font-weight:normal}
span.sansserif {font-family:sans-serif; font-weight:normal}
ul.no-bullet {list-style: none}
-->
</style>


</head>

<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="PLAIN-Monte-Carlo"></a>
<div class="header">
<p>
Next: <a href="MISER.html#MISER" accesskey="n" rel="next">MISER</a>, Previous: <a href="Monte-Carlo-Interface.html#Monte-Carlo-Interface" accesskey="p" rel="previous">Monte Carlo Interface</a>, Up: <a href="Monte-Carlo-Integration.html#Monte-Carlo-Integration" accesskey="u" rel="up">Monte Carlo Integration</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="PLAIN-Monte-Carlo-1"></a>
<h3 class="section">25.2 PLAIN Monte Carlo</h3>
<a name="index-plain-Monte-Carlo"></a>
<p>The plain Monte Carlo algorithm samples points randomly from the
integration region to estimate the integral and its error.  Using this
algorithm the estimate of the integral <em>E(f; N)</em> for <em>N</em>
randomly distributed points <em>x_i</em> is given by,
</p>
<div class="example">
<pre class="example">E(f; N) = =  V &lt;f&gt; = (V / N) \sum_i^N f(x_i)
</pre></div>

<p>where <em>V</em> is the volume of the integration region.  The error on
this estimate <em>\sigma(E;N)</em> is calculated from the estimated
variance of the mean,
</p>
<div class="example">
<pre class="example">\sigma^2 (E; N) = (V^2 / N^2) \sum_i^N (f(x_i) -  &lt;f&gt;)^2.
</pre></div>

<p>For large <em>N</em> this variance decreases asymptotically as
<em>\Var(f)/N</em>, where <em>\Var(f)</em> is the true variance of the
function over the integration region.  The error estimate itself should
decrease as <em>\sigma(f)/\sqrt{N}</em>.  The familiar law of errors
decreasing as <em>1/\sqrt{N}</em> applies&mdash;to reduce the error by a
factor of 10 requires a 100-fold increase in the number of sample
points.
</p>
<p>The functions described in this section are declared in the header file
<samp>gsl_monte_plain.h</samp>.
</p>
<dl>
<dt><a name="index-gsl_005fmonte_005fplain_005falloc"></a>Function: <em>gsl_monte_plain_state *</em> <strong>gsl_monte_plain_alloc</strong> <em>(size_t <var>dim</var>)</em></dt>
<dd><a name="index-gsl_005fmonte_005fplain_005fstate"></a>
<p>This function allocates and initializes a workspace for Monte Carlo
integration in <var>dim</var> dimensions.  
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fmonte_005fplain_005finit"></a>Function: <em>int</em> <strong>gsl_monte_plain_init</strong> <em>(gsl_monte_plain_state* <var>s</var>)</em></dt>
<dd><p>This function initializes a previously allocated integration state.
This allows an existing workspace to be reused for different
integrations.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fmonte_005fplain_005fintegrate"></a>Function: <em>int</em> <strong>gsl_monte_plain_integrate</strong> <em>(gsl_monte_function * <var>f</var>, const double <var>xl</var>[], const double <var>xu</var>[], size_t <var>dim</var>, size_t <var>calls</var>, gsl_rng * <var>r</var>, gsl_monte_plain_state * <var>s</var>, double * <var>result</var>, double * <var>abserr</var>)</em></dt>
<dd><p>This routines uses the plain Monte Carlo algorithm to integrate the
function <var>f</var> over the <var>dim</var>-dimensional hypercubic region
defined by the lower and upper limits in the arrays <var>xl</var> and
<var>xu</var>, each of size <var>dim</var>.  The integration uses a fixed number
of function calls <var>calls</var>, and obtains random sampling points using
the random number generator <var>r</var>. A previously allocated workspace
<var>s</var> must be supplied.  The result of the integration is returned in
<var>result</var>, with an estimated absolute error <var>abserr</var>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fmonte_005fplain_005ffree"></a>Function: <em>void</em> <strong>gsl_monte_plain_free</strong> <em>(gsl_monte_plain_state * <var>s</var>)</em></dt>
<dd><p>This function frees the memory associated with the integrator state
<var>s</var>.
</p></dd></dl>

<hr>
<div class="header">
<p>
Next: <a href="MISER.html#MISER" accesskey="n" rel="next">MISER</a>, Previous: <a href="Monte-Carlo-Interface.html#Monte-Carlo-Interface" accesskey="p" rel="previous">Monte Carlo Interface</a>, Up: <a href="Monte-Carlo-Integration.html#Monte-Carlo-Integration" accesskey="u" rel="up">Monte Carlo Integration</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>



</body>
</html>