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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
* 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 NPY_INFINITY
*/
/*
* Cephes Math Library Release 2.8: June, 2000
* Copyright 1984, 1987, 1992, 2000 by Stephen L. Moshier
*/
/*
* Code for the rational approximation on [1, 2] is:
*
* (C) Copyright John Maddock 2006.
* Use, modification and distribution are subject to the
* Boost Software License, Version 1.0. (See accompanying file
* LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
*/
#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
};
static double digamma_imp_1_2(double x)
{
/*
* Rational approximation on [1, 2] taken from Boost.
*
* Now for the approximation, we use the form:
*
* digamma(x) = (x - root) * (Y + R(x-1))
*
* Where root is the location of the positive root of digamma,
* Y is a constant, and R is optimised for low absolute error
* compared to Y.
*
* Maximum Deviation Found: 1.466e-18
* At double precision, max error found: 2.452e-17
*/
double r, g;
static const float Y = 0.99558162689208984f;
static const double root1 = 1569415565.0 / 1073741824.0;
static const double root2 = (381566830.0 / 1073741824.0) / 1073741824.0;
static const double root3 = 0.9016312093258695918615325266959189453125e-19;
static double P[] = {
-0.0020713321167745952,
-0.045251321448739056,
-0.28919126444774784,
-0.65031853770896507,
-0.32555031186804491,
0.25479851061131551
};
static double Q[] = {
-0.55789841321675513e-6,
0.0021284987017821144,
0.054151797245674225,
0.43593529692665969,
1.4606242909763515,
2.0767117023730469,
1.0
};
g = x - root1;
g -= root2;
g -= root3;
r = polevl(x - 1.0, P, 5) / polevl(x - 1.0, Q, 6);
return g * Y + g * r;
}
static double psi_asy(double x)
{
double y, z;
if (x < 1.0e17) {
z = 1.0 / (x * x);
y = z * polevl(z, A, 6);
}
else {
y = 0.0;
}
return log(x) - (0.5 / x) - y;
}
double psi(double x)
{
double y = 0.0;
double q, r;
int i, n;
if (npy_isnan(x)) {
return x;
}
else if (x == NPY_INFINITY) {
return x;
}
else if (x == -NPY_INFINITY) {
return NPY_NAN;
}
else if (x == 0) {
mtherr("psi", SING);
return npy_copysign(NPY_INFINITY, -x);
}
else if (x < 0.0) {
/* argument reduction before evaluating tan(pi * x) */
r = modf(x, &q);
if (r == 0.0) {
mtherr("psi", SING);
return NPY_NAN;
}
y = -NPY_PI / tan(NPY_PI * r);
x = 1.0 - x;
}
/* check for positive integer up to 10 */
if ((x <= 10.0) && (x == floor(x))) {
n = (int)x;
for (i = 1; i < n; i++) {
y += 1.0 / i;
}
y -= NPY_EULER;
return y;
}
/* use the recurrence relation to move x into [1, 2] */
if (x < 1.0) {
y -= 1.0 / x;
x += 1.0;
}
else if (x < 10.0) {
while (x > 2.0) {
x -= 1.0;
y += 1.0 / x;
}
}
if ((1.0 <= x) && (x <= 2.0)) {
y += digamma_imp_1_2(x);
return y;
}
/* x is large, use the asymptotic series */
y += psi_asy(x);
return y;
}
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