/* specfunc/hyperg_2F1.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000, 2004 Gerard Jungman
* Copyright (C) 2009 Brian Gough
*
* 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_exp.h>
#include <gsl/gsl_sf_pow_int.h>
#include <gsl/gsl_sf_gamma.h>
#include <gsl/gsl_sf_psi.h>
#include <gsl/gsl_sf_hyperg.h>
#include "error.h"
#define locEPS (1000.0*GSL_DBL_EPSILON)
/* Assumes c != negative integer.
*/
static int
hyperg_2F1_series(const double a, const double b, const double c,
const double x,
gsl_sf_result * result
)
{
double sum_pos = 1.0;
double sum_neg = 0.0;
double del_pos = 1.0;
double del_neg = 0.0;
double del = 1.0;
double del_prev;
double k = 0.0;
int i = 0;
if(fabs(c) < GSL_DBL_EPSILON) {
result->val = 0.0; /* FIXME: ?? */
result->err = 1.0;
GSL_ERROR ("error", GSL_EDOM);
}
do {
if(++i > 30000) {
result->val = sum_pos - sum_neg;
result->err = del_pos + del_neg;
result->err += 2.0 * GSL_DBL_EPSILON * (sum_pos + sum_neg);
result->err += 2.0 * GSL_DBL_EPSILON * (2.0*sqrt(k)+1.0) * fabs(result->val);
GSL_ERROR ("error", GSL_EMAXITER);
}
del_prev = del;
del *= (a+k)*(b+k) * x / ((c+k) * (k+1.0)); /* Gauss series */
if(del > 0.0) {
del_pos = del;
sum_pos += del;
}
else if(del == 0.0) {
/* Exact termination (a or b was a negative integer).
*/
del_pos = 0.0;
del_neg = 0.0;
break;
}
else {
del_neg = -del;
sum_neg -= del;
}
/*
* This stopping criteria is taken from the thesis
* "Computation of Hypergeometic Functions" by J. Pearson, pg. 31
* (http://people.maths.ox.ac.uk/porterm/research/pearson_final.pdf)
* and fixes bug #45926
*/
if (fabs(del_prev / (sum_pos - sum_neg)) < GSL_DBL_EPSILON &&
fabs(del / (sum_pos - sum_neg)) < GSL_DBL_EPSILON)
break;
k += 1.0;
} while(fabs((del_pos + del_neg)/(sum_pos-sum_neg)) > GSL_DBL_EPSILON);
result->val = sum_pos - sum_neg;
result->err = del_pos + del_neg;
result->err += 2.0 * GSL_DBL_EPSILON * (sum_pos + sum_neg);
result->err += 2.0 * GSL_DBL_EPSILON * (2.0*sqrt(k) + 1.0) * fabs(result->val);
return GSL_SUCCESS;
}
/* a = aR + i aI, b = aR - i aI */
static
int
hyperg_2F1_conj_series(const double aR, const double aI, const double c,
double x,
gsl_sf_result * result)
{
if(c == 0.0) {
result->val = 0.0; /* FIXME: should be Inf */
result->err = 0.0;
GSL_ERROR ("error", GSL_EDOM);
}
else {
double sum_pos = 1.0;
double sum_neg = 0.0;
double del_pos = 1.0;
double del_neg = 0.0;
double del = 1.0;
double k = 0.0;
do {
del *= ((aR+k)*(aR+k) + aI*aI)/((k+1.0)*(c+k)) * x;
if(del >= 0.0) {
del_pos = del;
sum_pos += del;
}
else {
del_neg = -del;
sum_neg -= del;
}
if(k > 30000) {
result->val = sum_pos - sum_neg;
result->err = del_pos + del_neg;
result->err += 2.0 * GSL_DBL_EPSILON * (sum_pos + sum_neg);
result->err += 2.0 * GSL_DBL_EPSILON * (2.0*sqrt(k)+1.0) * fabs(result->val);
GSL_ERROR ("error", GSL_EMAXITER);
}
k += 1.0;
} while(fabs((del_pos + del_neg)/(sum_pos - sum_neg)) > GSL_DBL_EPSILON);
result->val = sum_pos - sum_neg;
result->err = del_pos + del_neg;
result->err += 2.0 * GSL_DBL_EPSILON * (sum_pos + sum_neg);
result->err += 2.0 * GSL_DBL_EPSILON * (2.0*sqrt(k) + 1.0) * fabs(result->val);
return GSL_SUCCESS;
}
}
/* Luke's rational approximation. The most accesible
* discussion is in [Kolbig, CPC 23, 51 (1981)].
* The convergence is supposedly guaranteed for x < 0.
* You have to read Luke's books to see this and other
* results. Unfortunately, the stability is not so
* clear to me, although it seems very efficient when
* it works.
*/
static
int
hyperg_2F1_luke(const double a, const double b, const double c,
const double xin,
gsl_sf_result * result)
{
int stat_iter;
const double RECUR_BIG = 1.0e+50;
const int nmax = 20000;
int n = 3;
const double x = -xin;
const double x3 = x*x*x;
const double t0 = a*b/c;
const double t1 = (a+1.0)*(b+1.0)/(2.0*c);
const double t2 = (a+2.0)*(b+2.0)/(2.0*(c+1.0));
double F = 1.0;
double prec;
double Bnm3 = 1.0; /* B0 */
double Bnm2 = 1.0 + t1 * x; /* B1 */
double Bnm1 = 1.0 + t2 * x * (1.0 + t1/3.0 * x); /* B2 */
double Anm3 = 1.0; /* A0 */
double Anm2 = Bnm2 - t0 * x; /* A1 */
double Anm1 = Bnm1 - t0*(1.0 + t2*x)*x + t0 * t1 * (c/(c+1.0)) * x*x; /* A2 */
while(1) {
double npam1 = n + a - 1;
double npbm1 = n + b - 1;
double npcm1 = n + c - 1;
double npam2 = n + a - 2;
double npbm2 = n + b - 2;
double npcm2 = n + c - 2;
double tnm1 = 2*n - 1;
double tnm3 = 2*n - 3;
double tnm5 = 2*n - 5;
double n2 = n*n;
double F1 = (3.0*n2 + (a+b-6)*n + 2 - a*b - 2*(a+b)) / (2*tnm3*npcm1);
double F2 = -(3.0*n2 - (a+b+6)*n + 2 - a*b)*npam1*npbm1/(4*tnm1*tnm3*npcm2*npcm1);
double F3 = (npam2*npam1*npbm2*npbm1*(n-a-2)*(n-b-2)) / (8*tnm3*tnm3*tnm5*(n+c-3)*npcm2*npcm1);
double E = -npam1*npbm1*(n-c-1) / (2*tnm3*npcm2*npcm1);
double An = (1.0+F1*x)*Anm1 + (E + F2*x)*x*Anm2 + F3*x3*Anm3;
double Bn = (1.0+F1*x)*Bnm1 + (E + F2*x)*x*Bnm2 + F3*x3*Bnm3;
double r = An/Bn;
prec = fabs((F - r)/F);
F = r;
if(prec < GSL_DBL_EPSILON || n > nmax) break;
if(fabs(An) > RECUR_BIG || fabs(Bn) > RECUR_BIG) {
An /= RECUR_BIG;
Bn /= RECUR_BIG;
Anm1 /= RECUR_BIG;
Bnm1 /= RECUR_BIG;
Anm2 /= RECUR_BIG;
Bnm2 /= RECUR_BIG;
Anm3 /= RECUR_BIG;
Bnm3 /= RECUR_BIG;
}
else if(fabs(An) < 1.0/RECUR_BIG || fabs(Bn) < 1.0/RECUR_BIG) {
An *= RECUR_BIG;
Bn *= RECUR_BIG;
Anm1 *= RECUR_BIG;
Bnm1 *= RECUR_BIG;
Anm2 *= RECUR_BIG;
Bnm2 *= RECUR_BIG;
Anm3 *= RECUR_BIG;
Bnm3 *= RECUR_BIG;
}
n++;
Bnm3 = Bnm2;
Bnm2 = Bnm1;
Bnm1 = Bn;
Anm3 = Anm2;
Anm2 = Anm1;
Anm1 = An;
}
result->val = F;
result->err = 2.0 * fabs(prec * F);
result->err += 2.0 * GSL_DBL_EPSILON * (n+1.0) * fabs(F);
/* FIXME: just a hack: there's a lot of shit going on here */
result->err *= 8.0 * (fabs(a) + fabs(b) + 1.0);
stat_iter = (n >= nmax ? GSL_EMAXITER : GSL_SUCCESS );
return stat_iter;
}
/* Luke's rational approximation for the
* case a = aR + i aI, b = aR - i aI.
*/
static
int
hyperg_2F1_conj_luke(const double aR, const double aI, const double c,
const double xin,
gsl_sf_result * result)
{
int stat_iter;
const double RECUR_BIG = 1.0e+50;
const int nmax = 10000;
int n = 3;
const double x = -xin;
const double x3 = x*x*x;
const double atimesb = aR*aR + aI*aI;
const double apb = 2.0*aR;
const double t0 = atimesb/c;
const double t1 = (atimesb + apb + 1.0)/(2.0*c);
const double t2 = (atimesb + 2.0*apb + 4.0)/(2.0*(c+1.0));
double F = 1.0;
double prec;
double Bnm3 = 1.0; /* B0 */
double Bnm2 = 1.0 + t1 * x; /* B1 */
double Bnm1 = 1.0 + t2 * x * (1.0 + t1/3.0 * x); /* B2 */
double Anm3 = 1.0; /* A0 */
double Anm2 = Bnm2 - t0 * x; /* A1 */
double Anm1 = Bnm1 - t0*(1.0 + t2*x)*x + t0 * t1 * (c/(c+1.0)) * x*x; /* A2 */
while(1) {
double nm1 = n - 1;
double nm2 = n - 2;
double npam1_npbm1 = atimesb + nm1*apb + nm1*nm1;
double npam2_npbm2 = atimesb + nm2*apb + nm2*nm2;
double npcm1 = nm1 + c;
double npcm2 = nm2 + c;
double tnm1 = 2*n - 1;
double tnm3 = 2*n - 3;
double tnm5 = 2*n - 5;
double n2 = n*n;
double F1 = (3.0*n2 + (apb-6)*n + 2 - atimesb - 2*apb) / (2*tnm3*npcm1);
double F2 = -(3.0*n2 - (apb+6)*n + 2 - atimesb)*npam1_npbm1/(4*tnm1*tnm3*npcm2*npcm1);
double F3 = (npam2_npbm2*npam1_npbm1*(nm2*nm2 - nm2*apb + atimesb)) / (8*tnm3*tnm3*tnm5*(n+c-3)*npcm2*npcm1);
double E = -npam1_npbm1*(n-c-1) / (2*tnm3*npcm2*npcm1);
double An = (1.0+F1*x)*Anm1 + (E + F2*x)*x*Anm2 + F3*x3*Anm3;
double Bn = (1.0+F1*x)*Bnm1 + (E + F2*x)*x*Bnm2 + F3*x3*Bnm3;
double r = An/Bn;
prec = fabs(F - r)/fabs(F);
F = r;
if(prec < GSL_DBL_EPSILON || n > nmax) break;
if(fabs(An) > RECUR_BIG || fabs(Bn) > RECUR_BIG) {
An /= RECUR_BIG;
Bn /= RECUR_BIG;
Anm1 /= RECUR_BIG;
Bnm1 /= RECUR_BIG;
Anm2 /= RECUR_BIG;
Bnm2 /= RECUR_BIG;
Anm3 /= RECUR_BIG;
Bnm3 /= RECUR_BIG;
}
else if(fabs(An) < 1.0/RECUR_BIG || fabs(Bn) < 1.0/RECUR_BIG) {
An *= RECUR_BIG;
Bn *= RECUR_BIG;
Anm1 *= RECUR_BIG;
Bnm1 *= RECUR_BIG;
Anm2 *= RECUR_BIG;
Bnm2 *= RECUR_BIG;
Anm3 *= RECUR_BIG;
Bnm3 *= RECUR_BIG;
}
n++;
Bnm3 = Bnm2;
Bnm2 = Bnm1;
Bnm1 = Bn;
Anm3 = Anm2;
Anm2 = Anm1;
Anm1 = An;
}
result->val = F;
result->err = 2.0 * fabs(prec * F);
result->err += 2.0 * GSL_DBL_EPSILON * (n+1.0) * fabs(F);
/* FIXME: see above */
result->err *= 8.0 * (fabs(aR) + fabs(aI) + 1.0);
stat_iter = (n >= nmax ? GSL_EMAXITER : GSL_SUCCESS );
return stat_iter;
}
/* Do the reflection described in [Moshier, p. 334].
* Assumes a,b,c != neg integer.
*/
static
int
hyperg_2F1_reflect(const double a, const double b, const double c,
const double x, gsl_sf_result * result)
{
const double d = c - a - b;
const int intd = floor(d+0.5);
const int d_integer = ( fabs(d - intd) < locEPS );
if(d_integer) {
const double ln_omx = log(1.0 - x);
const double ad = fabs(d);
int stat_F2 = GSL_SUCCESS;
double sgn_2;
gsl_sf_result F1;
gsl_sf_result F2;
double d1, d2;
gsl_sf_result lng_c;
gsl_sf_result lng_ad2;
gsl_sf_result lng_bd2;
int stat_c;
int stat_ad2;
int stat_bd2;
if(d >= 0.0) {
d1 = d;
d2 = 0.0;
}
else {
d1 = 0.0;
d2 = d;
}
stat_ad2 = gsl_sf_lngamma_e(a+d2, &lng_ad2);
stat_bd2 = gsl_sf_lngamma_e(b+d2, &lng_bd2);
stat_c = gsl_sf_lngamma_e(c, &lng_c);
/* Evaluate F1.
*/
if(ad < GSL_DBL_EPSILON) {
/* d = 0 */
F1.val = 0.0;
F1.err = 0.0;
}
else {
gsl_sf_result lng_ad;
gsl_sf_result lng_ad1;
gsl_sf_result lng_bd1;
int stat_ad = gsl_sf_lngamma_e(ad, &lng_ad);
int stat_ad1 = gsl_sf_lngamma_e(a+d1, &lng_ad1);
int stat_bd1 = gsl_sf_lngamma_e(b+d1, &lng_bd1);
if(stat_ad1 == GSL_SUCCESS && stat_bd1 == GSL_SUCCESS && stat_ad == GSL_SUCCESS) {
/* Gamma functions in the denominator are ok.
* Proceed with evaluation.
*/
int i;
double sum1 = 1.0;
double term = 1.0;
double ln_pre1_val = lng_ad.val + lng_c.val + d2*ln_omx - lng_ad1.val - lng_bd1.val;
double ln_pre1_err = lng_ad.err + lng_c.err + lng_ad1.err + lng_bd1.err + GSL_DBL_EPSILON * fabs(ln_pre1_val);
int stat_e;
/* Do F1 sum.
*/
for(i=1; i<ad; i++) {
int j = i-1;
term *= (a + d2 + j) * (b + d2 + j) / (1.0 + d2 + j) / i * (1.0-x);
sum1 += term;
}
stat_e = gsl_sf_exp_mult_err_e(ln_pre1_val, ln_pre1_err,
sum1, GSL_DBL_EPSILON*fabs(sum1),
&F1);
if(stat_e == GSL_EOVRFLW) {
OVERFLOW_ERROR(result);
}
}
else {
/* Gamma functions in the denominator were not ok.
* So the F1 term is zero.
*/
F1.val = 0.0;
F1.err = 0.0;
}
} /* end F1 evaluation */
/* Evaluate F2.
*/
if(stat_ad2 == GSL_SUCCESS && stat_bd2 == GSL_SUCCESS) {
/* Gamma functions in the denominator are ok.
* Proceed with evaluation.
*/
const int maxiter = 2000;
double psi_1 = -M_EULER;
gsl_sf_result psi_1pd;
gsl_sf_result psi_apd1;
gsl_sf_result psi_bpd1;
int stat_1pd = gsl_sf_psi_e(1.0 + ad, &psi_1pd);
int stat_apd1 = gsl_sf_psi_e(a + d1, &psi_apd1);
int stat_bpd1 = gsl_sf_psi_e(b + d1, &psi_bpd1);
int stat_dall = GSL_ERROR_SELECT_3(stat_1pd, stat_apd1, stat_bpd1);
double psi_val = psi_1 + psi_1pd.val - psi_apd1.val - psi_bpd1.val - ln_omx;
double psi_err = psi_1pd.err + psi_apd1.err + psi_bpd1.err + GSL_DBL_EPSILON*fabs(psi_val);
double fact = 1.0;
double sum2_val = psi_val;
double sum2_err = psi_err;
double ln_pre2_val = lng_c.val + d1*ln_omx - lng_ad2.val - lng_bd2.val;
double ln_pre2_err = lng_c.err + lng_ad2.err + lng_bd2.err + GSL_DBL_EPSILON * fabs(ln_pre2_val);
int stat_e;
int j;
/* Do F2 sum.
*/
for(j=1; j<maxiter; j++) {
/* values for psi functions use recurrence; Abramowitz+Stegun 6.3.5 */
double term1 = 1.0/(double)j + 1.0/(ad+j);
double term2 = 1.0/(a+d1+j-1.0) + 1.0/(b+d1+j-1.0);
double delta = 0.0;
psi_val += term1 - term2;
psi_err += GSL_DBL_EPSILON * (fabs(term1) + fabs(term2));
fact *= (a+d1+j-1.0)*(b+d1+j-1.0)/((ad+j)*j) * (1.0-x);
delta = fact * psi_val;
sum2_val += delta;
sum2_err += fabs(fact * psi_err) + GSL_DBL_EPSILON*fabs(delta);
if(fabs(delta) < GSL_DBL_EPSILON * fabs(sum2_val)) break;
}
if(j == maxiter) stat_F2 = GSL_EMAXITER;
if(sum2_val == 0.0) {
F2.val = 0.0;
F2.err = 0.0;
}
else {
stat_e = gsl_sf_exp_mult_err_e(ln_pre2_val, ln_pre2_err,
sum2_val, sum2_err,
&F2);
if(stat_e == GSL_EOVRFLW) {
result->val = 0.0;
result->err = 0.0;
GSL_ERROR ("error", GSL_EOVRFLW);
}
}
stat_F2 = GSL_ERROR_SELECT_2(stat_F2, stat_dall);
}
else {
/* Gamma functions in the denominator not ok.
* So the F2 term is zero.
*/
F2.val = 0.0;
F2.err = 0.0;
} /* end F2 evaluation */
sgn_2 = ( GSL_IS_ODD(intd) ? -1.0 : 1.0 );
result->val = F1.val + sgn_2 * F2.val;
result->err = F1.err + F2. err;
result->err += 2.0 * GSL_DBL_EPSILON * (fabs(F1.val) + fabs(F2.val));
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return stat_F2;
}
else {
/* d not an integer */
gsl_sf_result pre1, pre2;
double sgn1, sgn2;
gsl_sf_result F1, F2;
int status_F1, status_F2;
/* These gamma functions appear in the denominator, so we
* catch their harmless domain errors and set the terms to zero.
*/
gsl_sf_result ln_g1ca, ln_g1cb, ln_g2a, ln_g2b;
double sgn_g1ca, sgn_g1cb, sgn_g2a, sgn_g2b;
int stat_1ca = gsl_sf_lngamma_sgn_e(c-a, &ln_g1ca, &sgn_g1ca);
int stat_1cb = gsl_sf_lngamma_sgn_e(c-b, &ln_g1cb, &sgn_g1cb);
int stat_2a = gsl_sf_lngamma_sgn_e(a, &ln_g2a, &sgn_g2a);
int stat_2b = gsl_sf_lngamma_sgn_e(b, &ln_g2b, &sgn_g2b);
int ok1 = (stat_1ca == GSL_SUCCESS && stat_1cb == GSL_SUCCESS);
int ok2 = (stat_2a == GSL_SUCCESS && stat_2b == GSL_SUCCESS);
gsl_sf_result ln_gc, ln_gd, ln_gmd;
double sgn_gc, sgn_gd, sgn_gmd;
gsl_sf_lngamma_sgn_e( c, &ln_gc, &sgn_gc);
gsl_sf_lngamma_sgn_e( d, &ln_gd, &sgn_gd);
gsl_sf_lngamma_sgn_e(-d, &ln_gmd, &sgn_gmd);
sgn1 = sgn_gc * sgn_gd * sgn_g1ca * sgn_g1cb;
sgn2 = sgn_gc * sgn_gmd * sgn_g2a * sgn_g2b;
if(ok1 && ok2) {
double ln_pre1_val = ln_gc.val + ln_gd.val - ln_g1ca.val - ln_g1cb.val;
double ln_pre2_val = ln_gc.val + ln_gmd.val - ln_g2a.val - ln_g2b.val + d*log(1.0-x);
double ln_pre1_err = ln_gc.err + ln_gd.err + ln_g1ca.err + ln_g1cb.err;
double ln_pre2_err = ln_gc.err + ln_gmd.err + ln_g2a.err + ln_g2b.err;
if(ln_pre1_val < GSL_LOG_DBL_MAX && ln_pre2_val < GSL_LOG_DBL_MAX) {
gsl_sf_exp_err_e(ln_pre1_val, ln_pre1_err, &pre1);
gsl_sf_exp_err_e(ln_pre2_val, ln_pre2_err, &pre2);
pre1.val *= sgn1;
pre2.val *= sgn2;
}
else {
OVERFLOW_ERROR(result);
}
}
else if(ok1 && !ok2) {
double ln_pre1_val = ln_gc.val + ln_gd.val - ln_g1ca.val - ln_g1cb.val;
double ln_pre1_err = ln_gc.err + ln_gd.err + ln_g1ca.err + ln_g1cb.err;
if(ln_pre1_val < GSL_LOG_DBL_MAX) {
gsl_sf_exp_err_e(ln_pre1_val, ln_pre1_err, &pre1);
pre1.val *= sgn1;
pre2.val = 0.0;
pre2.err = 0.0;
}
else {
OVERFLOW_ERROR(result);
}
}
else if(!ok1 && ok2) {
double ln_pre2_val = ln_gc.val + ln_gmd.val - ln_g2a.val - ln_g2b.val + d*log(1.0-x);
double ln_pre2_err = ln_gc.err + ln_gmd.err + ln_g2a.err + ln_g2b.err;
if(ln_pre2_val < GSL_LOG_DBL_MAX) {
pre1.val = 0.0;
pre1.err = 0.0;
gsl_sf_exp_err_e(ln_pre2_val, ln_pre2_err, &pre2);
pre2.val *= sgn2;
}
else {
OVERFLOW_ERROR(result);
}
}
else {
pre1.val = 0.0;
pre2.val = 0.0;
UNDERFLOW_ERROR(result);
}
status_F1 = hyperg_2F1_series( a, b, 1.0-d, 1.0-x, &F1);
status_F2 = hyperg_2F1_series(c-a, c-b, 1.0+d, 1.0-x, &F2);
result->val = pre1.val*F1.val + pre2.val*F2.val;
result->err = fabs(pre1.val*F1.err) + fabs(pre2.val*F2.err);
result->err += fabs(pre1.err*F1.val) + fabs(pre2.err*F2.val);
result->err += 2.0 * GSL_DBL_EPSILON * (fabs(pre1.val*F1.val) + fabs(pre2.val*F2.val));
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
if (status_F1)
return status_F1;
if (status_F2)
return status_F2;
return GSL_SUCCESS;
}
}
static int pow_omx(const double x, const double p, gsl_sf_result * result)
{
double ln_omx;
double ln_result;
if(fabs(x) < GSL_ROOT5_DBL_EPSILON) {
ln_omx = -x*(1.0 + x*(1.0/2.0 + x*(1.0/3.0 + x/4.0 + x*x/5.0)));
}
else {
ln_omx = log(1.0-x);
}
ln_result = p * ln_omx;
return gsl_sf_exp_err_e(ln_result, GSL_DBL_EPSILON * fabs(ln_result), result);
}
/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
int
gsl_sf_hyperg_2F1_e(double a, double b, const double c,
const double x,
gsl_sf_result * result)
{
const double d = c - a - b;
const double rinta = floor(a + 0.5);
const double rintb = floor(b + 0.5);
const double rintc = floor(c + 0.5);
const int a_neg_integer = ( a < 0.0 && fabs(a - rinta) < locEPS );
const int b_neg_integer = ( b < 0.0 && fabs(b - rintb) < locEPS );
const int c_neg_integer = ( c < 0.0 && fabs(c - rintc) < locEPS );
result->val = 0.0;
result->err = 0.0;
/* Handle x == 1.0 RJM */
if (fabs (x - 1.0) < locEPS && (c - a - b) > 0 && c != 0 && !c_neg_integer) {
gsl_sf_result lngamc, lngamcab, lngamca, lngamcb;
double lngamc_sgn, lngamca_sgn, lngamcb_sgn;
int status;
int stat1 = gsl_sf_lngamma_sgn_e (c, &lngamc, &lngamc_sgn);
int stat2 = gsl_sf_lngamma_e (c - a - b, &lngamcab);
int stat3 = gsl_sf_lngamma_sgn_e (c - a, &lngamca, &lngamca_sgn);
int stat4 = gsl_sf_lngamma_sgn_e (c - b, &lngamcb, &lngamcb_sgn);
if (stat1 != GSL_SUCCESS || stat2 != GSL_SUCCESS
|| stat3 != GSL_SUCCESS || stat4 != GSL_SUCCESS)
{
DOMAIN_ERROR (result);
}
status =
gsl_sf_exp_err_e (lngamc.val + lngamcab.val - lngamca.val - lngamcb.val,
lngamc.err + lngamcab.err + lngamca.err + lngamcb.err,
result);
result->val *= lngamc_sgn / (lngamca_sgn * lngamcb_sgn);
return status;
}
if(x < -1.0 || 1.0 <= x) {
DOMAIN_ERROR(result);
}
if(c_neg_integer) {
/* If c is a negative integer, then either a or b must be a
negative integer of smaller magnitude than c to ensure
cancellation of the series. */
if(! (a_neg_integer && a > c + 0.1) && ! (b_neg_integer && b > c + 0.1)) {
DOMAIN_ERROR(result);
}
}
if(fabs(c-b) < locEPS || fabs(c-a) < locEPS) {
return pow_omx(x, d, result); /* (1-x)^(c-a-b) */
}
if(a >= 0.0 && b >= 0.0 && c >=0.0 && x >= 0.0 && x < 0.995) {
/* Series has all positive definite
* terms and x is not close to 1.
*/
return hyperg_2F1_series(a, b, c, x, result);
}
if(fabs(a) < 10.0 && fabs(b) < 10.0) {
/* a and b are not too large, so we attempt
* variations on the series summation.
*/
if(a_neg_integer) {
return hyperg_2F1_series(rinta, b, c, x, result);
}
if(b_neg_integer) {
return hyperg_2F1_series(a, rintb, c, x, result);
}
if(x < -0.25) {
return hyperg_2F1_luke(a, b, c, x, result);
}
else if(x < 0.5) {
return hyperg_2F1_series(a, b, c, x, result);
}
else {
if(fabs(c) > 10.0) {
return hyperg_2F1_series(a, b, c, x, result);
}
else {
return hyperg_2F1_reflect(a, b, c, x, result);
}
}
}
else {
/* Either a or b or both large.
* Introduce some new variables ap,bp so that bp is
* the larger in magnitude.
*/
double ap, bp;
if(fabs(a) > fabs(b)) {
bp = a;
ap = b;
}
else {
bp = b;
ap = a;
}
if(x < 0.0) {
/* What the hell, maybe Luke will converge.
*/
return hyperg_2F1_luke(a, b, c, x, result);
}
if(GSL_MAX_DBL(fabs(ap),1.0)*fabs(bp)*fabs(x) < 2.0*fabs(c)) {
/* If c is large enough or x is small enough,
* we can attempt the series anyway.
*/
return hyperg_2F1_series(a, b, c, x, result);
}
if(fabs(bp*bp*x*x) < 0.001*fabs(bp) && fabs(ap) < 10.0) {
/* The famous but nearly worthless "large b" asymptotic.
*/
int stat = gsl_sf_hyperg_1F1_e(ap, c, bp*x, result);
result->err = 0.001 * fabs(result->val);
return stat;
}
/* We give up. */
result->val = 0.0;
result->err = 0.0;
GSL_ERROR ("error", GSL_EUNIMPL);
}
}
int
gsl_sf_hyperg_2F1_conj_e(const double aR, const double aI, const double c,
const double x,
gsl_sf_result * result)
{
const double ax = fabs(x);
const double rintc = floor(c + 0.5);
const int c_neg_integer = ( c < 0.0 && fabs(c - rintc) < locEPS );
result->val = 0.0;
result->err = 0.0;
if(ax >= 1.0 || c_neg_integer || c == 0.0) {
DOMAIN_ERROR(result);
}
if( (ax < 0.25 && fabs(aR) < 20.0 && fabs(aI) < 20.0)
|| (c > 0.0 && x > 0.0)
) {
return hyperg_2F1_conj_series(aR, aI, c, x, result);
}
else if(fabs(aR) < 10.0 && fabs(aI) < 10.0) {
if(x < -0.25) {
return hyperg_2F1_conj_luke(aR, aI, c, x, result);
}
else {
return hyperg_2F1_conj_series(aR, aI, c, x, result);
}
}
else {
if(x < 0.0) {
/* What the hell, maybe Luke will converge.
*/
return hyperg_2F1_conj_luke(aR, aI, c, x, result);
}
/* Give up. */
result->val = 0.0;
result->err = 0.0;
GSL_ERROR ("error", GSL_EUNIMPL);
}
}
int
gsl_sf_hyperg_2F1_renorm_e(const double a, const double b, const double c,
const double x,
gsl_sf_result * result
)
{
const double rinta = floor(a + 0.5);
const double rintb = floor(b + 0.5);
const double rintc = floor(c + 0.5);
const int a_neg_integer = ( a < 0.0 && fabs(a - rinta) < locEPS );
const int b_neg_integer = ( b < 0.0 && fabs(b - rintb) < locEPS );
const int c_neg_integer = ( c < 0.0 && fabs(c - rintc) < locEPS );
if(c_neg_integer) {
if((a_neg_integer && a > c+0.1) || (b_neg_integer && b > c+0.1)) {
/* 2F1 terminates early */
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else {
/* 2F1 does not terminate early enough, so something survives */
/* [Abramowitz+Stegun, 15.1.2] */
gsl_sf_result g1, g2, g3, g4, g5;
double s1, s2, s3, s4, s5;
int stat = 0;
stat += gsl_sf_lngamma_sgn_e(a-c+1, &g1, &s1);
stat += gsl_sf_lngamma_sgn_e(b-c+1, &g2, &s2);
stat += gsl_sf_lngamma_sgn_e(a, &g3, &s3);
stat += gsl_sf_lngamma_sgn_e(b, &g4, &s4);
stat += gsl_sf_lngamma_sgn_e(-c+2, &g5, &s5);
if(stat != 0) {
DOMAIN_ERROR(result);
}
else {
gsl_sf_result F;
int stat_F = gsl_sf_hyperg_2F1_e(a-c+1, b-c+1, -c+2, x, &F);
double ln_pre_val = g1.val + g2.val - g3.val - g4.val - g5.val;
double ln_pre_err = g1.err + g2.err + g3.err + g4.err + g5.err;
double sg = s1 * s2 * s3 * s4 * s5;
int stat_e = gsl_sf_exp_mult_err_e(ln_pre_val, ln_pre_err,
sg * F.val, F.err,
result);
return GSL_ERROR_SELECT_2(stat_e, stat_F);
}
}
}
else {
/* generic c */
gsl_sf_result F;
gsl_sf_result lng;
double sgn;
int stat_g = gsl_sf_lngamma_sgn_e(c, &lng, &sgn);
int stat_F = gsl_sf_hyperg_2F1_e(a, b, c, x, &F);
int stat_e = gsl_sf_exp_mult_err_e(-lng.val, lng.err,
sgn*F.val, F.err,
result);
return GSL_ERROR_SELECT_3(stat_e, stat_F, stat_g);
}
}
int
gsl_sf_hyperg_2F1_conj_renorm_e(const double aR, const double aI, const double c,
const double x,
gsl_sf_result * result
)
{
const double rintc = floor(c + 0.5);
const double rinta = floor(aR + 0.5);
const int a_neg_integer = ( aR < 0.0 && fabs(aR-rinta) < locEPS && aI == 0.0);
const int c_neg_integer = ( c < 0.0 && fabs(c - rintc) < locEPS );
if(c_neg_integer) {
if(a_neg_integer && aR > c+0.1) {
/* 2F1 terminates early */
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else {
/* 2F1 does not terminate early enough, so something survives */
/* [Abramowitz+Stegun, 15.1.2] */
gsl_sf_result g1, g2;
gsl_sf_result g3;
gsl_sf_result a1, a2;
int stat = 0;
stat += gsl_sf_lngamma_complex_e(aR-c+1, aI, &g1, &a1);
stat += gsl_sf_lngamma_complex_e(aR, aI, &g2, &a2);
stat += gsl_sf_lngamma_e(-c+2.0, &g3);
if(stat != 0) {
DOMAIN_ERROR(result);
}
else {
gsl_sf_result F;
int stat_F = gsl_sf_hyperg_2F1_conj_e(aR-c+1, aI, -c+2, x, &F);
double ln_pre_val = 2.0*(g1.val - g2.val) - g3.val;
double ln_pre_err = 2.0 * (g1.err + g2.err) + g3.err;
int stat_e = gsl_sf_exp_mult_err_e(ln_pre_val, ln_pre_err,
F.val, F.err,
result);
return GSL_ERROR_SELECT_2(stat_e, stat_F);
}
}
}
else {
/* generic c */
gsl_sf_result F;
gsl_sf_result lng;
double sgn;
int stat_g = gsl_sf_lngamma_sgn_e(c, &lng, &sgn);
int stat_F = gsl_sf_hyperg_2F1_conj_e(aR, aI, c, x, &F);
int stat_e = gsl_sf_exp_mult_err_e(-lng.val, lng.err,
sgn*F.val, F.err,
result);
return GSL_ERROR_SELECT_3(stat_e, stat_F, stat_g);
}
}
/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
#include "eval.h"
double gsl_sf_hyperg_2F1(double a, double b, double c, double x)
{
EVAL_RESULT(gsl_sf_hyperg_2F1_e(a, b, c, x, &result));
}
double gsl_sf_hyperg_2F1_conj(double aR, double aI, double c, double x)
{
EVAL_RESULT(gsl_sf_hyperg_2F1_conj_e(aR, aI, c, x, &result));
}
double gsl_sf_hyperg_2F1_renorm(double a, double b, double c, double x)
{
EVAL_RESULT(gsl_sf_hyperg_2F1_renorm_e(a, b, c, x, &result));
}
double gsl_sf_hyperg_2F1_conj_renorm(double aR, double aI, double c, double x)
{
EVAL_RESULT(gsl_sf_hyperg_2F1_conj_renorm_e(aR, aI, c, x, &result));
}