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
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
|
/* beta.c
*
* Beta function
*
*
*
* SYNOPSIS:
*
* double a, b, y, beta();
*
* y = beta( a, b );
*
*
*
* DESCRIPTION:
*
* - -
* | (a) | (b)
* beta( a, b ) = -----------.
* -
* | (a+b)
*
* For large arguments the logarithm of the function is
* evaluated using lgam(), then exponentiated.
*
*
*
* ACCURACY:
*
* Relative error:
* arithmetic domain # trials peak rms
* DEC 0,30 1700 7.7e-15 1.5e-15
* IEEE 0,30 30000 8.1e-14 1.1e-14
*
* ERROR MESSAGES:
*
* message condition value returned
* beta overflow log(beta) > MAXLOG 0.0
* a or b <0 integer 0.0
*
*/
/* beta.c */
/*
* Cephes Math Library Release 2.0: April, 1987
* Copyright 1984, 1987 by Stephen L. Moshier
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
#include "mconf.h"
#ifdef UNK
#define MAXGAM 34.84425627277176174
#endif
#ifdef DEC
#define MAXGAM 34.84425627277176174
#endif
#ifdef IBMPC
#define MAXGAM 171.624376956302725
#endif
#ifdef MIEEE
#define MAXGAM 171.624376956302725
#endif
extern double MAXLOG;
extern int sgngam;
#define ASYMP_FACTOR 1e6
static double lbeta_asymp(double a, double b, int *sgn);
static double lbeta_negint(int a, double b);
static double beta_negint(int a, double b);
double gammasgn(double x);
double beta(a, b)
double a, b;
{
double y;
int sign;
sign = 1;
if (a <= 0.0) {
if (a == floor(a)) {
if (a == (int)a) {
return beta_negint((int)a, b);
}
else {
goto over;
}
}
}
if (b <= 0.0) {
if (b == floor(b)) {
if (b == (int)b) {
return beta_negint((int)b, a);
}
else {
goto over;
}
}
}
if (fabs(a) < fabs(b)) {
y = a; a = b; b = y;
}
if (fabs(a) > ASYMP_FACTOR * fabs(b) && a > ASYMP_FACTOR) {
/* Avoid loss of precision in lgam(a + b) - lgam(a) */
y = lbeta_asymp(a, b, &sign);
return sign * exp(y);
}
y = a + b;
if (fabs(y) > MAXGAM || fabs(a) > MAXGAM || fabs(b) > MAXGAM) {
y = lgam(y);
sign *= sgngam; /* keep track of the sign */
y = lgam(b) - y;
sign *= sgngam;
y = lgam(a) + y;
sign *= sgngam;
if (y > MAXLOG) {
over:
mtherr("beta", OVERFLOW);
return (sign * NPY_INFINITY);
}
return (sign * exp(y));
}
y = Gamma(y);
if (y == 0.0)
goto over;
if (a > b) {
y = Gamma(a) / y;
y *= Gamma(b);
}
else {
y = Gamma(b) / y;
y *= Gamma(a);
}
return (y);
}
/* Natural log of |beta|. Return the sign of beta in sgngam. */
double lbeta(a, b)
double a, b;
{
double y;
int sign;
sign = 1;
if (a <= 0.0) {
if (a == floor(a)) {
if (a == (int)a) {
return lbeta_negint((int)a, b);
}
else {
goto over;
}
}
}
if (b <= 0.0) {
if (b == floor(b)) {
if (b == (int)b) {
return lbeta_negint((int)b, a);
}
else {
goto over;
}
}
}
if (fabs(a) < fabs(b)) {
y = a; a = b; b = y;
}
if (fabs(a) > ASYMP_FACTOR * fabs(b) && a > ASYMP_FACTOR) {
/* Avoid loss of precision in lgam(a + b) - lgam(a) */
y = lbeta_asymp(a, b, &sign);
sgngam = sign;
return y;
}
y = a + b;
if (fabs(y) > MAXGAM || fabs(a) > MAXGAM || fabs(b) > MAXGAM) {
y = lgam(y);
sign *= sgngam; /* keep track of the sign */
y = lgam(b) - y;
sign *= sgngam;
y = lgam(a) + y;
sign *= sgngam;
sgngam = sign;
return (y);
}
y = Gamma(y);
if (y == 0.0) {
over:
mtherr("lbeta", OVERFLOW);
return (sign * NPY_INFINITY);
}
if (a > b) {
y = Gamma(a) / y;
y *= Gamma(b);
}
else {
y = Gamma(b) / y;
y *= Gamma(a);
}
if (y < 0) {
sgngam = -1;
y = -y;
}
else
sgngam = 1;
return (log(y));
}
/*
* Asymptotic expansion for ln(|B(a, b)|) for a > ASYMP_FACTOR*max(|b|, 1).
*/
static double lbeta_asymp(double a, double b, int *sgn)
{
double r, sum;
r = lgam(b);
*sgn = sgngam;
r -= b * log(a);
r += b*(1-b)/(2*a);
r += b*(1-b)*(1-2*b)/(12*a*a);
r += - b*b*(1-b)*(1-b)/(12*a*a*a);
return r;
}
/*
* Special case for a negative integer argument
*/
static double beta_negint(int a, double b)
{
int sgn;
if (b == (int)b && 1 - a - b > 0) {
sgn = ((int)b % 2 == 0) ? 1 : -1;
return sgn * beta(1 - a - b, b);
}
else {
mtherr("lbeta", OVERFLOW);
return NPY_INFINITY;
}
}
static double lbeta_negint(int a, double b)
{
double r;
int sgn;
if (b == (int)b && 1 - a - b > 0) {
sgn = ((int)b % 2 == 0) ? 1 : -1;
r = lbeta(1 - a - b, b);
sgngam *= sgn;
return r;
}
else {
mtherr("lbeta", OVERFLOW);
return NPY_INFINITY;
}
}
|
