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
|
<html lang="en">
<head>
<title>QAWF adaptive integration for Fourier integrals - GNU Scientific Library -- Reference Manual</title>
<meta http-equiv="Content-Type" content="text/html">
<meta name="description" content="GNU Scientific Library -- Reference Manual">
<meta name="generator" content="makeinfo 4.8">
<link title="Top" rel="start" href="index.html#Top">
<link rel="up" href="Numerical-Integration.html" title="Numerical Integration">
<link rel="prev" href="QAWO-adaptive-integration-for-oscillatory-functions.html" title="QAWO adaptive integration for oscillatory functions">
<link rel="next" href="Numerical-integration-error-codes.html" title="Numerical integration error codes">
<link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage">
<!--
Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007 The GSL Team.
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.2 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 freedom to copy and modify this
GNU Manual, like GNU software.''-->
<meta http-equiv="Content-Style-Type" content="text/css">
<style type="text/css"><!--
pre.display { font-family:inherit }
pre.format { font-family:inherit }
pre.smalldisplay { font-family:inherit; font-size:smaller }
pre.smallformat { font-family:inherit; font-size:smaller }
pre.smallexample { font-size:smaller }
pre.smalllisp { font-size:smaller }
span.sc { font-variant:small-caps }
span.roman { font-family:serif; font-weight:normal; }
span.sansserif { font-family:sans-serif; font-weight:normal; }
--></style>
</head>
<body>
<div class="node">
<p>
<a name="QAWF-adaptive-integration-for-Fourier-integrals"></a>
Next: <a rel="next" accesskey="n" href="Numerical-integration-error-codes.html">Numerical integration error codes</a>,
Previous: <a rel="previous" accesskey="p" href="QAWO-adaptive-integration-for-oscillatory-functions.html">QAWO adaptive integration for oscillatory functions</a>,
Up: <a rel="up" accesskey="u" href="Numerical-Integration.html">Numerical Integration</a>
<hr>
</div>
<h3 class="section">16.10 QAWF adaptive integration for Fourier integrals</h3>
<p><a name="index-QAWF-quadrature-algorithm-1474"></a><a name="index-Fourier-integrals_002c-numerical-1475"></a>
<div class="defun">
— Function: int <b>gsl_integration_qawf</b> (<var>gsl_function * f, const double a, const double epsabs, const size_t limit, gsl_integration_workspace * workspace, gsl_integration_workspace * cycle_workspace, gsl_integration_qawo_table * wf, double * result, double * abserr</var>)<var><a name="index-gsl_005fintegration_005fqawf-1476"></a></var><br>
<blockquote>
<p>This function attempts to compute a Fourier integral of the function
<var>f</var> over the semi-infinite interval [a,+\infty).
<pre class="example"> I = \int_a^{+\infty} dx f(x) sin(omega x)
I = \int_a^{+\infty} dx f(x) cos(omega x)
</pre>
<p>The parameter \omega and choice of \sin or \cos is
taken from the table <var>wf</var> (the length <var>L</var> can take any value,
since it is overridden by this function to a value appropriate for the
fourier integration). The integral is computed using the QAWO algorithm
over each of the subintervals,
<pre class="example"> C_1 = [a, a + c]
C_2 = [a + c, a + 2 c]
... = ...
C_k = [a + (k-1) c, a + k c]
</pre>
<p class="noindent">where
<!-- {$c = (2 \,\hbox{floor}(|\omega|) + 1) \pi/|\omega|$} -->
c = (2 floor(|\omega|) + 1) \pi/|\omega|. The width c is
chosen to cover an odd number of periods so that the contributions from
the intervals alternate in sign and are monotonically decreasing when
<var>f</var> is positive and monotonically decreasing. The sum of this
sequence of contributions is accelerated using the epsilon-algorithm.
<p>This function works to an overall absolute tolerance of
<var>abserr</var>. The following strategy is used: on each interval
C_k the algorithm tries to achieve the tolerance
<pre class="example"> TOL_k = u_k abserr
</pre>
<p class="noindent">where
<!-- {$u_k = (1 - p)p^{k-1}$} -->
u_k = (1 - p)p^{k-1} and p = 9/10.
The sum of the geometric series of contributions from each interval
gives an overall tolerance of <var>abserr</var>.
<p>If the integration of a subinterval leads to difficulties then the
accuracy requirement for subsequent intervals is relaxed,
<pre class="example"> TOL_k = u_k max(abserr, max_{i<k}{E_i})
</pre>
<p class="noindent">where E_k is the estimated error on the interval C_k.
<p>The subintervals and their results are stored in the memory provided by
<var>workspace</var>. The maximum number of subintervals is given by
<var>limit</var>, which may not exceed the allocated size of the workspace.
The integration over each subinterval uses the memory provided by
<var>cycle_workspace</var> as workspace for the QAWO algorithm.
</blockquote></div>
<hr>The GNU Scientific Library - a free numerical library licensed under the GNU GPL<br>Back to the <a href="/software/gsl/">GNU Scientific Library Homepage</a></body></html>
|
