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
|
/* psi.c
*
* Psi (digamma) function
*
*
* SYNOPSIS:
*
* double x, y, psi();
*
* y = psi( x );
*
*
* DESCRIPTION:
*
* d -
* psi(x) = -- ln | (x)
* dx
*
* is the logarithmic derivative of the gamma function.
* For integer x,
* n-1
* -
* psi(n) = -EUL + > 1/k.
* -
* k=1
*
* This formula is used for 0 < n <= 10. If x is negative, it
* is transformed to a positive argument by the reflection
* formula psi(1-x) = psi(x) + pi cot(pi x).
* For general positive x, the argument is made greater than 10
* using the recurrence psi(x+1) = psi(x) + 1/x.
* Then the following asymptotic expansion is applied:
*
* inf. B
* - 2k
* psi(x) = log(x) - 1/2x - > -------
* - 2k
* k=1 2k x
*
* where the B2k are Bernoulli numbers.
*
* ACCURACY:
* Relative error (except absolute when |psi| < 1):
* arithmetic domain # trials peak rms
* DEC 0,30 2500 1.7e-16 2.0e-17
* IEEE 0,30 30000 1.3e-15 1.4e-16
* IEEE -30,0 40000 1.5e-15 2.2e-16
*
* ERROR MESSAGES:
* message condition value returned
* psi singularity x integer <=0 MAXNUM
*/
/*
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1992, 2000 by Stephen L. Moshier
*/
#include <math.h>
#ifdef UNK
static double A[] = {
8.33333333333333333333E-2,
-2.10927960927960927961E-2,
7.57575757575757575758E-3,
-4.16666666666666666667E-3,
3.96825396825396825397E-3,
-8.33333333333333333333E-3,
8.33333333333333333333E-2
};
#endif
#ifdef DEC
static unsigned short A[] = {
0037252,0125252,0125252,0125253,
0136654,0145314,0126312,0146255,
0036370,0037017,0101740,0174076,
0136210,0104210,0104210,0104211,
0036202,0004040,0101010,0020202,
0136410,0104210,0104210,0104211,
0037252,0125252,0125252,0125253
};
#endif
#ifdef IBMPC
static unsigned short A[] = {
0x5555,0x5555,0x5555,0x3fb5,
0x5996,0x9599,0x9959,0xbf95,
0x1f08,0xf07c,0x07c1,0x3f7f,
0x1111,0x1111,0x1111,0xbf71,
0x0410,0x1041,0x4104,0x3f70,
0x1111,0x1111,0x1111,0xbf81,
0x5555,0x5555,0x5555,0x3fb5
};
#endif
#ifdef MIEEE
static unsigned short A[] = {
0x3fb5,0x5555,0x5555,0x5555,
0xbf95,0x9959,0x9599,0x5996,
0x3f7f,0x07c1,0xf07c,0x1f08,
0xbf71,0x1111,0x1111,0x1111,
0x3f70,0x4104,0x1041,0x0410,
0xbf81,0x1111,0x1111,0x1111,
0x3fb5,0x5555,0x5555,0x5555
};
#endif
#define EUL 0.57721566490153286061
#ifdef ANSIPROT
extern double floor ( double );
extern double log ( double );
extern double tan ( double );
extern double polevl ( double, void *, int );
#else
double floor(), log(), tan(), polevl();
#endif
extern double PI, MAXNUM;
double psi(x)
double x;
{
double p, q, nz, s, w, y, z;
int i, n, negative;
negative = 0;
nz = 0.0;
if( x <= 0.0 )
{
negative = 1;
q = x;
p = floor(q);
if( p == q )
{
mtherr( "psi", SING );
return( MAXNUM );
}
/* Remove the zeros of tan(PI x)
* by subtracting the nearest integer from x
*/
nz = q - p;
if( nz != 0.5 )
{
if( nz > 0.5 )
{
p += 1.0;
nz = q - p;
}
nz = PI/tan(PI*nz);
}
else
{
nz = 0.0;
}
x = 1.0 - x;
}
/* check for positive integer up to 10 */
if( (x <= 10.0) && (x == floor(x)) )
{
y = 0.0;
n = x;
for( i=1; i<n; i++ )
{
w = i;
y += 1.0/w;
}
y -= EUL;
goto done;
}
s = x;
w = 0.0;
while( s < 10.0 )
{
w += 1.0/s;
s += 1.0;
}
if( s < 1.0e17 )
{
z = 1.0/(s * s);
y = z * polevl( z, A, 6 );
}
else
y = 0.0;
y = log(s) - (0.5/s) - y - w;
done:
if( negative )
{
y -= nz;
}
return(y);
}
|
