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
|
<html lang="en">
<head>
<title>Example of accelerating a series - 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="Series-Acceleration.html" title="Series Acceleration">
<link rel="prev" href="Acceleration-functions-without-error-estimation.html" title="Acceleration functions without error estimation">
<link rel="next" href="Series-Acceleration-References.html" title="Series Acceleration References">
<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="Example-of-accelerating-a-series"></a>
Next: <a rel="next" accesskey="n" href="Series-Acceleration-References.html">Series Acceleration References</a>,
Previous: <a rel="previous" accesskey="p" href="Acceleration-functions-without-error-estimation.html">Acceleration functions without error estimation</a>,
Up: <a rel="up" accesskey="u" href="Series-Acceleration.html">Series Acceleration</a>
<hr>
</div>
<h3 class="section">29.3 Examples</h3>
<p>The following code calculates an estimate of \zeta(2) = \pi^2 / 6
using the series,
<pre class="example"> \zeta(2) = 1 + 1/2^2 + 1/3^2 + 1/4^2 + ...
</pre>
<p class="noindent">After <var>N</var> terms the error in the sum is O(1/N), making direct
summation of the series converge slowly.
<pre class="example"><pre class="verbatim"> #include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_sum.h>
#define N 20
int
main (void)
{
double t[N];
double sum_accel, err;
double sum = 0;
int n;
gsl_sum_levin_u_workspace * w
= gsl_sum_levin_u_alloc (N);
const double zeta_2 = M_PI * M_PI / 6.0;
/* terms for zeta(2) = \sum_{n=1}^{\infty} 1/n^2 */
for (n = 0; n < N; n++)
{
double np1 = n + 1.0;
t[n] = 1.0 / (np1 * np1);
sum += t[n];
}
gsl_sum_levin_u_accel (t, N, w, &sum_accel, &err);
printf ("term-by-term sum = % .16f using %d terms\n",
sum, N);
printf ("term-by-term sum = % .16f using %d terms\n",
w->sum_plain, w->terms_used);
printf ("exact value = % .16f\n", zeta_2);
printf ("accelerated sum = % .16f using %d terms\n",
sum_accel, w->terms_used);
printf ("estimated error = % .16f\n", err);
printf ("actual error = % .16f\n",
sum_accel - zeta_2);
gsl_sum_levin_u_free (w);
return 0;
}
</pre></pre>
<p class="noindent">The output below shows that the Levin u-transform is able to obtain an
estimate of the sum to 1 part in
<!-- {$10^{10}$} -->
10^10 using the first eleven terms of the series. The
error estimate returned by the function is also accurate, giving
the correct number of significant digits.
<pre class="example"> $ ./a.out
<pre class="verbatim"> term-by-term sum = 1.5961632439130233 using 20 terms
term-by-term sum = 1.5759958390005426 using 13 terms
exact value = 1.6449340668482264
accelerated sum = 1.6449340668166479 using 13 terms
estimated error = 0.0000000000508580
actual error = -0.0000000000315785
</pre></pre>
<p class="noindent">Note that a direct summation of this series would require
<!-- {$10^{10}$} -->
10^10 terms to achieve the same precision as the accelerated
sum does in 13 terms.
<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>
|
