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/* 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 "mconf.h"
static double A[] = {
8.33333333333333333333E-2,
-2.10927960927960927961E-2,
7.57575757575757575758E-3,
-4.16666666666666666667E-3,
3.96825396825396825397E-3,
-8.33333333333333333333E-3,
8.33333333333333333333E-2
};
#define EUL 0.57721566490153286061
double psi (double x)
{
double p, q, nz, s, w, y, z;
int i, n, negative = 0;
nz = 0.0;
if (x <= 0.0) {
negative = 1;
q = x;
p = floor(q);
if (p == q) {
mtherr_with_arg("psi", CEPHES_SING, x);
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;
}
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