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
155
|
/* @(#)w_gamma.c 5.1 93/09/24 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*
*/
/* BUG: FIXME?
According to Linux man pages for tgamma, lgamma, and gamma, the gamma
function was originally defined in BSD as implemented here--the log of the gamma
function. BSD 4.3 changed the name to lgamma, apparently removing gamma. BSD
4.4 re-introduced the gamma name with the more intuitive, without logarithm,
plain gamma function. The C99 standard apparently wanted to avoid a problem
with the poorly-named earlier gamma and used tgamma when adding a plain
gamma function.
So the current gamma is matching an old, bad definition, and not
matching a newer, better definition. */
/*
FUNCTION
<<gamma>>, <<gammaf>>, <<lgamma>>, <<lgammaf>>, <<gamma_r>>, <<gammaf_r>>, <<lgamma_r>>, <<lgammaf_r>>, <<tgamma>>, and <<tgammaf>>---logarithmic and plain gamma functions
INDEX
gamma
INDEX
gammaf
INDEX
lgamma
INDEX
lgammaf
INDEX
gamma_r
INDEX
gammaf_r
INDEX
lgamma_r
INDEX
lgammaf_r
INDEX
tgamma
INDEX
tgammaf
SYNOPSIS
#include <math.h>
double gamma(double <[x]>);
float gammaf(float <[x]>);
double lgamma(double <[x]>);
float lgammaf(float <[x]>);
double gamma_r(double <[x]>, int *<[signgamp]>);
float gammaf_r(float <[x]>, int *<[signgamp]>);
double lgamma_r(double <[x]>, int *<[signgamp]>);
float lgammaf_r(float <[x]>, int *<[signgamp]>);
double tgamma(double <[x]>);
float tgammaf(float <[x]>);
DESCRIPTION
<<gamma>> calculates
@tex
$\mit ln\bigl(\Gamma(x)\bigr)$,
@end tex
the natural logarithm of the gamma function of <[x]>. The gamma function
(<<exp(gamma(<[x]>))>>) is a generalization of factorial, and retains
the property that
@ifnottex
<<exp(gamma(N))>> is equivalent to <<N*exp(gamma(N-1))>>.
@end ifnottex
@tex
$\mit \Gamma(N)\equiv N\times\Gamma(N-1)$.
@end tex
Accordingly, the results of the gamma function itself grow very
quickly. <<gamma>> is defined as
@tex
$\mit ln\bigl(\Gamma(x)\bigr)$ rather than simply $\mit \Gamma(x)$
@end tex
@ifnottex
the natural log of the gamma function, rather than the gamma function
itself,
@end ifnottex
to extend the useful range of results representable.
The sign of the result is returned in the global variable <<signgam>>,
which is declared in math.h.
<<gammaf>> performs the same calculation as <<gamma>>, but uses and
returns <<float>> values.
<<lgamma>> and <<lgammaf>> are alternate names for <<gamma>> and
<<gammaf>>. The use of <<lgamma>> instead of <<gamma>> is a reminder
that these functions compute the log of the gamma function, rather
than the gamma function itself.
The functions <<gamma_r>>, <<gammaf_r>>, <<lgamma_r>>, and
<<lgammaf_r>> are just like <<gamma>>, <<gammaf>>, <<lgamma>>, and
<<lgammaf>>, respectively, but take an additional argument. This
additional argument is a pointer to an integer. This additional
argument is used to return the sign of the result, and the global
variable <<signgam>> is not used. These functions may be used for
reentrant calls (but they will still set the global variable <<errno>>
if an error occurs).
<<tgamma>> and <<tgammaf>> are the "true gamma" functions, returning
@tex
$\mit \Gamma(x)$,
@end tex
the gamma function of <[x]>--without a logarithm.
(They are apparently so named because of the prior existence of the old,
poorly-named <<gamma>> functions which returned the log of gamma up
through BSD 4.2.)
RETURNS
Normally, the computed result is returned.
When <[x]> is a nonpositive integer, <<gamma>> returns <<HUGE_VAL>>
and <<errno>> is set to <<EDOM>>. If the result overflows, <<gamma>>
returns <<HUGE_VAL>> and <<errno>> is set to <<ERANGE>>.
PORTABILITY
Neither <<gamma>> nor <<gammaf>> is ANSI C. It is better not to use either
of these; use <<lgamma>> or <<tgamma>> instead.@*
<<lgamma>>, <<lgammaf>>, <<tgamma>>, and <<tgammaf>> are nominally C standard
in terms of the base return values, although the <[signgam]> global for
<<lgamma>> is not standard.
*/
/* double gamma(double x)
* Return the logarithm of the Gamma function of x.
*
* Method: call lgamma
*/
#include "fdlibm.h"
#include <errno.h>
#ifndef _DOUBLE_IS_32BITS
#if !defined(_IEEE_LIBM) || !defined(HAVE_ALIAS_ATTRIBUTE)
#ifdef __STDC__
double gamma(double x)
#else
double gamma(x)
double x;
#endif
{
return lgamma(x);
}
#endif
#endif /* defined(_DOUBLE_IS_32BITS) */
|
