/* 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 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(

)

{

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));

}