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/* 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
* 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
*
*/
/*
* 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"
#define MAXGAM 171.624376956302725
extern double MAXLOG;
#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 beta(double a, double b)
{
double y;
int sign = 1;
if (a <= 0.0) {
if (a == floor(a)) {
if (a == (int)a) {
return beta_negint((int)a, b);
}
else {
goto overflow;
}
}
}
if (b <= 0.0) {
if (b == floor(b)) {
if (b == (int)b) {
return beta_negint((int)b, a);
}
else {
goto overflow;
}
}
}
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) {
int sgngam;
y = lgam_sgn(y, &sgngam);
sign *= sgngam; /* keep track of the sign */
y = lgam_sgn(b, &sgngam) - y;
sign *= sgngam;
y = lgam_sgn(a, &sgngam) + y;
sign *= sgngam;
if (y > MAXLOG) {
goto overflow;
}
return (sign * exp(y));
}
y = Gamma(y);
a = Gamma(a);
b = Gamma(b);
if (y == 0.0)
goto overflow;
if (fabs(fabs(a) - fabs(y)) > fabs(fabs(b) - fabs(y))) {
y = b / y;
y *= a;
}
else {
y = a / y;
y *= b;
}
return (y);
overflow:
mtherr("beta", OVERFLOW);
return (sign * NPY_INFINITY);
}
/* Natural log of |beta|. */
double lbeta(double a, double 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);
return y;
}
y = a + b;
if (fabs(y) > MAXGAM || fabs(a) > MAXGAM || fabs(b) > MAXGAM) {
int sgngam;
y = lgam_sgn(y, &sgngam);
sign *= sgngam; /* keep track of the sign */
y = lgam_sgn(b, &sgngam) - y;
sign *= sgngam;
y = lgam_sgn(a, &sgngam) + y;
sign *= sgngam;
return (y);
}
y = Gamma(y);
a = Gamma(a);
b = Gamma(b);
if (y == 0.0) {
over:
mtherr("lbeta", OVERFLOW);
return (sign * NPY_INFINITY);
}
if (fabs(fabs(a) - fabs(y)) > fabs(fabs(b) - fabs(y))) {
y = b / y;
y *= a;
}
else {
y = a / y;
y *= b;
}
if (y < 0) {
y = -y;
}
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 = lgam_sgn(b, sgn);
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;
if (b == (int)b && 1 - a - b > 0) {
r = lbeta(1 - a - b, b);
return r;
}
else {
mtherr("lbeta", OVERFLOW);
return NPY_INFINITY;
}
}
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