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
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
|
// Special functions -*- C++ -*-
// Copyright (C) 2006-2022 Free Software Foundation, Inc.
//
// This file is part of the GNU ISO C++ Library. This library is free
// software; you can redistribute it and/or modify it under the
// terms of the GNU General Public License as published by the
// Free Software Foundation; either version 3, or (at your option)
// any later version.
//
// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
// GNU General Public License for more details.
//
// Under Section 7 of GPL version 3, you are granted additional
// permissions described in the GCC Runtime Library Exception, version
// 3.1, as published by the Free Software Foundation.
// You should have received a copy of the GNU General Public License and
// a copy of the GCC Runtime Library Exception along with this program;
// see the files COPYING3 and COPYING.RUNTIME respectively. If not, see
// <http://www.gnu.org/licenses/>.
/** @file tr1/beta_function.tcc
* This is an internal header file, included by other library headers.
* Do not attempt to use it directly. @headername{tr1/cmath}
*/
//
// ISO C++ 14882 TR1: 5.2 Special functions
//
// Written by Edward Smith-Rowland based on:
// (1) Handbook of Mathematical Functions,
// ed. Milton Abramowitz and Irene A. Stegun,
// Dover Publications,
// Section 6, pp. 253-266
// (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl
// (3) Numerical Recipes in C, by W. H. Press, S. A. Teukolsky,
// W. T. Vetterling, B. P. Flannery, Cambridge University Press (1992),
// 2nd ed, pp. 213-216
// (4) Gamma, Exploring Euler's Constant, Julian Havil,
// Princeton, 2003.
#ifndef _GLIBCXX_TR1_BETA_FUNCTION_TCC
#define _GLIBCXX_TR1_BETA_FUNCTION_TCC 1
namespace std _GLIBCXX_VISIBILITY(default)
{
_GLIBCXX_BEGIN_NAMESPACE_VERSION
#if _GLIBCXX_USE_STD_SPEC_FUNCS
# define _GLIBCXX_MATH_NS ::std
#elif defined(_GLIBCXX_TR1_CMATH)
namespace tr1
{
# define _GLIBCXX_MATH_NS ::std::tr1
#else
# error do not include this header directly, use <cmath> or <tr1/cmath>
#endif
// [5.2] Special functions
// Implementation-space details.
namespace __detail
{
/**
* @brief Return the beta function: \f$B(x,y)\f$.
*
* The beta function is defined by
* @f[
* B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
* @f]
*
* @param __x The first argument of the beta function.
* @param __y The second argument of the beta function.
* @return The beta function.
*/
template<typename _Tp>
_Tp
__beta_gamma(_Tp __x, _Tp __y)
{
_Tp __bet;
#if _GLIBCXX_USE_C99_MATH_TR1
if (__x > __y)
{
__bet = _GLIBCXX_MATH_NS::tgamma(__x)
/ _GLIBCXX_MATH_NS::tgamma(__x + __y);
__bet *= _GLIBCXX_MATH_NS::tgamma(__y);
}
else
{
__bet = _GLIBCXX_MATH_NS::tgamma(__y)
/ _GLIBCXX_MATH_NS::tgamma(__x + __y);
__bet *= _GLIBCXX_MATH_NS::tgamma(__x);
}
#else
if (__x > __y)
{
__bet = __gamma(__x) / __gamma(__x + __y);
__bet *= __gamma(__y);
}
else
{
__bet = __gamma(__y) / __gamma(__x + __y);
__bet *= __gamma(__x);
}
#endif
return __bet;
}
/**
* @brief Return the beta function \f$B(x,y)\f$ using
* the log gamma functions.
*
* The beta function is defined by
* @f[
* B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
* @f]
*
* @param __x The first argument of the beta function.
* @param __y The second argument of the beta function.
* @return The beta function.
*/
template<typename _Tp>
_Tp
__beta_lgamma(_Tp __x, _Tp __y)
{
#if _GLIBCXX_USE_C99_MATH_TR1
_Tp __bet = _GLIBCXX_MATH_NS::lgamma(__x)
+ _GLIBCXX_MATH_NS::lgamma(__y)
- _GLIBCXX_MATH_NS::lgamma(__x + __y);
#else
_Tp __bet = __log_gamma(__x)
+ __log_gamma(__y)
- __log_gamma(__x + __y);
#endif
__bet = std::exp(__bet);
return __bet;
}
/**
* @brief Return the beta function \f$B(x,y)\f$ using
* the product form.
*
* The beta function is defined by
* @f[
* B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
* @f]
*
* @param __x The first argument of the beta function.
* @param __y The second argument of the beta function.
* @return The beta function.
*/
template<typename _Tp>
_Tp
__beta_product(_Tp __x, _Tp __y)
{
_Tp __bet = (__x + __y) / (__x * __y);
unsigned int __max_iter = 1000000;
for (unsigned int __k = 1; __k < __max_iter; ++__k)
{
_Tp __term = (_Tp(1) + (__x + __y) / __k)
/ ((_Tp(1) + __x / __k) * (_Tp(1) + __y / __k));
__bet *= __term;
}
return __bet;
}
/**
* @brief Return the beta function \f$ B(x,y) \f$.
*
* The beta function is defined by
* @f[
* B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}
* @f]
*
* @param __x The first argument of the beta function.
* @param __y The second argument of the beta function.
* @return The beta function.
*/
template<typename _Tp>
inline _Tp
__beta(_Tp __x, _Tp __y)
{
if (__isnan(__x) || __isnan(__y))
return std::numeric_limits<_Tp>::quiet_NaN();
else
return __beta_lgamma(__x, __y);
}
} // namespace __detail
#undef _GLIBCXX_MATH_NS
#if ! _GLIBCXX_USE_STD_SPEC_FUNCS && defined(_GLIBCXX_TR1_CMATH)
} // namespace tr1
#endif
_GLIBCXX_END_NAMESPACE_VERSION
}
#endif // _GLIBCXX_TR1_BETA_FUNCTION_TCC
|
