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+/* specfunc/hyperg_2F1.c
+ *
+ * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2004 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 2 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 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 *= (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;
+ }
+
+ 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);
+
+ 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;
+
+ if(x < -1.0 || 1.0 <= x) {
+ DOMAIN_ERROR(result);
+ }
+
+ if(c_neg_integer) {
+ if(! (a_neg_integer && a > c + 0.1)) DOMAIN_ERROR(result);
+ if(! (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(a),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(a) < 10.0) {
+ /* The famous but nearly worthless "large b" asymptotic.
+ */
+ int stat = gsl_sf_hyperg_1F1_e(a, 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));
+}
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