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
143
144
145
146
147
148
149
150
151
152
153
154
|
<!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 – Reference Manual: Factorials</title>
<meta name="description" content="GNU Scientific Library – Reference Manual: Factorials">
<meta name="keywords" content="GNU Scientific Library – Reference Manual: Factorials">
<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="Gamma-and-Beta-Functions.html#Gamma-and-Beta-Functions" rel="up" title="Gamma and Beta Functions">
<link href="Pochhammer-Symbol.html#Pochhammer-Symbol" rel="next" title="Pochhammer Symbol">
<link href="Gamma-Functions.html#Gamma-Functions" rel="previous" title="Gamma Functions">
<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="Factorials"></a>
<div class="header">
<p>
Next: <a href="Pochhammer-Symbol.html#Pochhammer-Symbol" accesskey="n" rel="next">Pochhammer Symbol</a>, Previous: <a href="Gamma-Functions.html#Gamma-Functions" accesskey="p" rel="previous">Gamma Functions</a>, Up: <a href="Gamma-and-Beta-Functions.html#Gamma-and-Beta-Functions" accesskey="u" rel="up">Gamma and Beta Functions</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Factorials-1"></a>
<h4 class="subsection">7.19.2 Factorials</h4>
<a name="index-factorial"></a>
<p>Although factorials can be computed from the Gamma function, using
the relation <em>n! = \Gamma(n+1)</em> for non-negative integer <em>n</em>,
it is usually more efficient to call the functions in this section,
particularly for small values of <em>n</em>, whose factorial values are
maintained in hardcoded tables.
</p>
<dl>
<dt><a name="index-gsl_005fsf_005ffact"></a>Function: <em>double</em> <strong>gsl_sf_fact</strong> <em>(unsigned int <var>n</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005ffact_005fe"></a>Function: <em>int</em> <strong>gsl_sf_fact_e</strong> <em>(unsigned int <var>n</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><a name="index-factorial-1"></a>
<p>These routines compute the factorial <em>n!</em>. The factorial is
related to the Gamma function by <em>n! = \Gamma(n+1)</em>.
The maximum value of <em>n</em> such that <em>n!</em> is not
considered an overflow is given by the macro <code>GSL_SF_FACT_NMAX</code>
and is 170.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fsf_005fdoublefact"></a>Function: <em>double</em> <strong>gsl_sf_doublefact</strong> <em>(unsigned int <var>n</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fdoublefact_005fe"></a>Function: <em>int</em> <strong>gsl_sf_doublefact_e</strong> <em>(unsigned int <var>n</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><a name="index-double-factorial"></a>
<p>These routines compute the double factorial <em>n!! = n(n-2)(n-4) \dots</em>.
The maximum value of <em>n</em> such that <em>n!!</em> is not
considered an overflow is given by the macro <code>GSL_SF_DOUBLEFACT_NMAX</code>
and is 297.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fsf_005flnfact"></a>Function: <em>double</em> <strong>gsl_sf_lnfact</strong> <em>(unsigned int <var>n</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005flnfact_005fe"></a>Function: <em>int</em> <strong>gsl_sf_lnfact_e</strong> <em>(unsigned int <var>n</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><a name="index-logarithm-of-factorial"></a>
<p>These routines compute the logarithm of the factorial of <var>n</var>,
<em>\log(n!)</em>. The algorithm is faster than computing
<em>\ln(\Gamma(n+1))</em> via <code>gsl_sf_lngamma</code> for <em>n < 170</em>,
but defers for larger <var>n</var>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fsf_005flndoublefact"></a>Function: <em>double</em> <strong>gsl_sf_lndoublefact</strong> <em>(unsigned int <var>n</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005flndoublefact_005fe"></a>Function: <em>int</em> <strong>gsl_sf_lndoublefact_e</strong> <em>(unsigned int <var>n</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><a name="index-logarithm-of-double-factorial"></a>
<p>These routines compute the logarithm of the double factorial of <var>n</var>,
<em>\log(n!!)</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fsf_005fchoose"></a>Function: <em>double</em> <strong>gsl_sf_choose</strong> <em>(unsigned int <var>n</var>, unsigned int <var>m</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005fchoose_005fe"></a>Function: <em>int</em> <strong>gsl_sf_choose_e</strong> <em>(unsigned int <var>n</var>, unsigned int <var>m</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><a name="index-combinatorial-factor-C_0028m_002cn_0029"></a>
<p>These routines compute the combinatorial factor <code>n choose m</code>
<em>= n!/(m!(n-m)!)</em>
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fsf_005flnchoose"></a>Function: <em>double</em> <strong>gsl_sf_lnchoose</strong> <em>(unsigned int <var>n</var>, unsigned int <var>m</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005flnchoose_005fe"></a>Function: <em>int</em> <strong>gsl_sf_lnchoose_e</strong> <em>(unsigned int <var>n</var>, unsigned int <var>m</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><a name="index-logarithm-of-combinatorial-factor-C_0028m_002cn_0029"></a>
<p>These routines compute the logarithm of <code>n choose m</code>. This is
equivalent to the sum <em>\log(n!) - \log(m!) - \log((n-m)!)</em>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fsf_005ftaylorcoeff"></a>Function: <em>double</em> <strong>gsl_sf_taylorcoeff</strong> <em>(int <var>n</var>, double <var>x</var>)</em></dt>
<dt><a name="index-gsl_005fsf_005ftaylorcoeff_005fe"></a>Function: <em>int</em> <strong>gsl_sf_taylorcoeff_e</strong> <em>(int <var>n</var>, double <var>x</var>, gsl_sf_result * <var>result</var>)</em></dt>
<dd><a name="index-Taylor-coefficients_002c-computation-of"></a>
<p>These routines compute the Taylor coefficient <em>x^n / n!</em> for
<em>x >= 0</em>,
<em>n >= 0</em>.
</p></dd></dl>
<hr>
<div class="header">
<p>
Next: <a href="Pochhammer-Symbol.html#Pochhammer-Symbol" accesskey="n" rel="next">Pochhammer Symbol</a>, Previous: <a href="Gamma-Functions.html#Gamma-Functions" accesskey="p" rel="previous">Gamma Functions</a>, Up: <a href="Gamma-and-Beta-Functions.html#Gamma-and-Beta-Functions" accesskey="u" rel="up">Gamma and Beta Functions</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
</body>
</html>
|
