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/* specfunc/beta_inc.c
*
* Copyright (C) 2007 Brian Gough
* 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 */
#include <config.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_sf_log.h>
#include <gsl/gsl_sf_exp.h>
#include <gsl/gsl_sf_gamma.h>
#include <gsl/gsl_sf_hyperg.h>
#include "error.h"
#include "check.h"
static double
isnegint (const double x)
{
return (x < 0) && (x == floor(x));
}
static
int
beta_cont_frac(
const double a,
const double b,
const double x,
gsl_sf_result * result
)
{
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 = cutoff;
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 = cutoff;
if(fabs(num_term) < cutoff) num_term = cutoff;
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 = cutoff;
if(fabs(num_term) < cutoff) num_term = cutoff;
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;
++iter_count;
}
result->val = cf;
result->err = iter_count * 4.0 * GSL_DBL_EPSILON * fabs(cf);
if(iter_count >= max_iter)
GSL_ERROR ("error", GSL_EMAXITER);
else
return GSL_SUCCESS;
}
/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
int
gsl_sf_beta_inc_e(
const double a,
const double b,
const double x,
gsl_sf_result * result
)
{
if(x < 0.0 || x > 1.0) {
DOMAIN_ERROR(result);
} else if (isnegint(a) || isnegint(b)) {
DOMAIN_ERROR(result);
} else if (isnegint(a+b)) {
DOMAIN_ERROR(result);
} else if(x == 0.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(x == 1.0) {
result->val = 1.0;
result->err = 0.0;
return GSL_SUCCESS;
} else if (a <= 0 || b <= 0) {
gsl_sf_result f, beta;
int stat;
const int stat_f = gsl_sf_hyperg_2F1_e(a, 1-b, a+1, x, &f);
const int stat_beta = gsl_sf_beta_e(a, b, &beta);
double prefactor = (pow(x, a) / a);
result->val = prefactor * f.val / beta.val;
result->err = fabs(prefactor) * f.err/ fabs(beta.val) + fabs(result->val/beta.val) * beta.err;
stat = GSL_ERROR_SELECT_2(stat_f, stat_beta);
if(stat == GSL_SUCCESS) {
CHECK_UNDERFLOW(result);
}
return stat;
} else {
gsl_sf_result ln_beta;
gsl_sf_result ln_x;
gsl_sf_result ln_1mx;
gsl_sf_result prefactor;
const int stat_ln_beta = gsl_sf_lnbeta_e(a, b, &ln_beta);
const int stat_ln_1mx = gsl_sf_log_1plusx_e(-x, &ln_1mx);
const int stat_ln_x = gsl_sf_log_e(x, &ln_x);
const int stat_ln = GSL_ERROR_SELECT_3(stat_ln_beta, stat_ln_1mx, stat_ln_x);
const double ln_pre_val = -ln_beta.val + a * ln_x.val + b * ln_1mx.val;
const double ln_pre_err = ln_beta.err + fabs(a*ln_x.err) + fabs(b*ln_1mx.err);
const int stat_exp = gsl_sf_exp_err_e(ln_pre_val, ln_pre_err, &prefactor);
if(stat_ln != GSL_SUCCESS) {
result->val = 0.0;
result->err = 0.0;
GSL_ERROR ("error", GSL_ESANITY);
}
if(x < (a + 1.0)/(a+b+2.0)) {
/* Apply continued fraction directly. */
gsl_sf_result cf;
const int stat_cf = beta_cont_frac(a, b, x, &cf);
int stat;
result->val = prefactor.val * cf.val / a;
result->err = (fabs(prefactor.err * cf.val) + fabs(prefactor.val * cf.err))/a;
stat = GSL_ERROR_SELECT_2(stat_exp, stat_cf);
if(stat == GSL_SUCCESS) {
CHECK_UNDERFLOW(result);
}
return stat;
}
else {
/* Apply continued fraction after hypergeometric transformation. */
gsl_sf_result cf;
const int stat_cf = beta_cont_frac(b, a, 1.0-x, &cf);
int stat;
const double term = prefactor.val * cf.val / b;
result->val = 1.0 - term;
result->err = fabs(prefactor.err * cf.val)/b;
result->err += fabs(prefactor.val * cf.err)/b;
result->err += 2.0 * GSL_DBL_EPSILON * (1.0 + fabs(term));
/* since the prefactor term is subtracted from 1 we need to
ignore underflow */
if (stat_exp != GSL_EUNDRFLW) {
stat = GSL_ERROR_SELECT_2(stat_exp, stat_cf);
} else {
stat = stat_cf;
};
if(stat == GSL_SUCCESS) {
CHECK_UNDERFLOW(result);
}
return stat;
}
}
}
/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
#include "eval.h"
double gsl_sf_beta_inc(const double a, const double b, const double x)
{
EVAL_RESULT(gsl_sf_beta_inc_e(a, b, x, &result));
}
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