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/* specfunc/beta_inc.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 3 of the License, or (at
* your option) any later version.
*
* This program is distributed in the hope that it will be useful, but
* WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/
/* Author: G. Jungman */
/* Modified for cdfs by Brian Gough, June 2003 */
#include <gsl/gsl_sf_gamma.h>
static double
beta_cont_frac (const double a, const double b, const double x,
const double epsabs)
{
const unsigned int max_iter = 512; /* control iterations */
const double cutoff = 2.0 * GSL_DBL_MIN; /* control the zero cutoff */
unsigned int iter_count = 0;
double cf;
/* standard initialization for continued fraction */
double num_term = 1.0;
double den_term = 1.0 - (a + b) * x / (a + 1.0);
if (fabs (den_term) < cutoff)
den_term = GSL_NAN;
den_term = 1.0 / den_term;
cf = den_term;
while (iter_count < max_iter)
{
const int k = iter_count + 1;
double coeff = k * (b - k) * x / (((a - 1.0) + 2 * k) * (a + 2 * k));
double delta_frac;
/* first step */
den_term = 1.0 + coeff * den_term;
num_term = 1.0 + coeff / num_term;
if (fabs (den_term) < cutoff)
den_term = GSL_NAN;
if (fabs (num_term) < cutoff)
num_term = GSL_NAN;
den_term = 1.0 / den_term;
delta_frac = den_term * num_term;
cf *= delta_frac;
coeff = -(a + k) * (a + b + k) * x / ((a + 2 * k) * (a + 2 * k + 1.0));
/* second step */
den_term = 1.0 + coeff * den_term;
num_term = 1.0 + coeff / num_term;
if (fabs (den_term) < cutoff)
den_term = GSL_NAN;
if (fabs (num_term) < cutoff)
num_term = GSL_NAN;
den_term = 1.0 / den_term;
delta_frac = den_term * num_term;
cf *= delta_frac;
if (fabs (delta_frac - 1.0) < 2.0 * GSL_DBL_EPSILON)
break;
if (cf * fabs (delta_frac - 1.0) < epsabs)
break;
++iter_count;
}
if (iter_count >= max_iter)
return GSL_NAN;
return cf;
}
/* The function beta_inc_AXPY(A,Y,a,b,x) computes A * beta_inc(a,b,x)
+ Y taking account of possible cancellations when using the
hypergeometric transformation beta_inc(a,b,x)=1-beta_inc(b,a,1-x).
It also adjusts the accuracy of beta_inc() to fit the overall
absolute error when A*beta_inc is added to Y. (e.g. if Y >>
A*beta_inc then the accuracy of beta_inc can be reduced) */
static double
beta_inc_AXPY (const double A, const double Y,
const double a, const double b, const double x)
{
if (x == 0.0)
{
return A * 0 + Y;
}
else if (x == 1.0)
{
return A * 1 + Y;
}
else if (a > 1e5 && b < 10 && x > a / (a + b))
{
/* Handle asymptotic regime, large a, small b, x > peak [AS 26.5.17] */
double N = a + (b - 1.0) / 2.0;
return A * gsl_sf_gamma_inc_Q (b, -N * log (x)) + Y;
}
else if (b > 1e5 && a < 10 && x < b / (a + b))
{
/* Handle asymptotic regime, small a, large b, x < peak [AS 26.5.17] */
double N = b + (a - 1.0) / 2.0;
return A * gsl_sf_gamma_inc_P (a, -N * log1p (-x)) + Y;
}
else
{
double ln_beta = gsl_sf_lnbeta (a, b);
double ln_pre = -ln_beta + a * log (x) + b * log1p (-x);
double prefactor = exp (ln_pre);
if (x < (a + 1.0) / (a + b + 2.0))
{
/* Apply continued fraction directly. */
double epsabs = fabs (Y / (A * prefactor / a)) * GSL_DBL_EPSILON;
double cf = beta_cont_frac (a, b, x, epsabs);
return A * (prefactor * cf / a) + Y;
}
else
{
/* Apply continued fraction after hypergeometric transformation. */
double epsabs =
fabs ((A + Y) / (A * prefactor / b)) * GSL_DBL_EPSILON;
double cf = beta_cont_frac (b, a, 1.0 - x, epsabs);
double term = prefactor * cf / b;
if (A == -Y)
{
return -A * term;
}
else
{
return A * (1 - term) + Y;
}
}
}
}
/* Direct series evaluation for testing purposes only */
#if 0
static double
beta_series (const double a, const double b, const double x,
const double epsabs)
{
double f = x / (1 - x);
double c = (b - 1) / (a + 1) * f;
double s = 1;
double n = 0;
s += c;
do
{
n++;
c *= -f * (2 + n - b) / (2 + n + a);
s += c;
}
while (n < 512 && fabs (c) > GSL_DBL_EPSILON * fabs (s) + epsabs);
s /= (1 - x);
return s;
}
#endif
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