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+/* specfunc/hyperg_U.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 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_gamma.h>
+#include <gsl/gsl_sf_bessel.h>
+#include <gsl/gsl_sf_laguerre.h>
+#include <gsl/gsl_sf_pow_int.h>
+#include <gsl/gsl_sf_hyperg.h>
+
+#include "error.h"
+#include "hyperg.h"
+
+#define INT_THRESHOLD (1000.0*GSL_DBL_EPSILON)
+
+#define SERIES_EVAL_OK(a,b,x) ((fabs(a) < 5 && b < 5 && x < 2.0) || (fabs(a) < 10 && b < 10 && x < 1.0))
+
+#define ASYMP_EVAL_OK(a,b,x) (GSL_MAX_DBL(fabs(a),1.0)*GSL_MAX_DBL(fabs(1.0+a-b),1.0) < 0.99*fabs(x))
+
+
+/* Log[U(a,2a,x)]
+ * [Abramowitz+stegun, 13.6.21]
+ * Assumes x > 0, a > 1/2.
+ */
+static
+int
+hyperg_lnU_beq2a(const double a, const double x, gsl_sf_result * result)
+{
+ const double lx = log(x);
+ const double nu = a - 0.5;
+ const double lnpre = 0.5*(x - M_LNPI) - nu*lx;
+ gsl_sf_result lnK;
+ gsl_sf_bessel_lnKnu_e(nu, 0.5*x, &lnK);
+ result->val = lnpre + lnK.val;
+ result->err = 2.0 * GSL_DBL_EPSILON * (fabs(0.5*x) + 0.5*M_LNPI + fabs(nu*lx));
+ result->err += lnK.err;
+ result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
+ return GSL_SUCCESS;
+}
+
+
+/* Evaluate u_{N+1}/u_N by Steed's continued fraction method.
+ *
+ * u_N := Gamma[a+N]/Gamma[a] U(a + N, b, x)
+ *
+ * u_{N+1}/u_N = (a+N) U(a+N+1,b,x)/U(a+N,b,x)
+ */
+static
+int
+hyperg_U_CF1(const double a, const double b, const int N, const double x,
+ double * result, int * count)
+{
+ const double RECUR_BIG = GSL_SQRT_DBL_MAX;
+ const int maxiter = 20000;
+ int n = 1;
+ double Anm2 = 1.0;
+ double Bnm2 = 0.0;
+ double Anm1 = 0.0;
+ double Bnm1 = 1.0;
+ double a1 = -(a + N);
+ double b1 = (b - 2.0*a - x - 2.0*(N+1));
+ double An = b1*Anm1 + a1*Anm2;
+ double Bn = b1*Bnm1 + a1*Bnm2;
+ double an, bn;
+ double fn = An/Bn;
+
+ while(n < maxiter) {
+ double old_fn;
+ double del;
+ n++;
+ Anm2 = Anm1;
+ Bnm2 = Bnm1;
+ Anm1 = An;
+ Bnm1 = Bn;
+ an = -(a + N + n - b)*(a + N + n - 1.0);
+ bn = (b - 2.0*a - x - 2.0*(N+n));
+ An = bn*Anm1 + an*Anm2;
+ Bn = bn*Bnm1 + an*Bnm2;
+
+ 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;
+ }
+
+ old_fn = fn;
+ fn = An/Bn;
+ del = old_fn/fn;
+
+ if(fabs(del - 1.0) < 10.0*GSL_DBL_EPSILON) break;
+ }
+
+ *result = fn;
+ *count = n;
+
+ if(n == maxiter)
+ GSL_ERROR ("error", GSL_EMAXITER);
+ else
+ return GSL_SUCCESS;
+}
+
+
+/* Large x asymptotic for x^a U(a,b,x)
+ * Based on SLATEC D9CHU() [W. Fullerton]
+ *
+ * Uses a rational approximation due to Luke.
+ * See [Luke, Algorithms for the Computation of Special Functions, p. 252]
+ * [Luke, Utilitas Math. (1977)]
+ *
+ * z^a U(a,b,z) ~ 2F0(a,1+a-b,-1/z)
+ *
+ * This assumes that a is not a negative integer and
+ * that 1+a-b is not a negative integer. If one of them
+ * is, then the 2F0 actually terminates, the above
+ * relation is an equality, and the sum should be
+ * evaluated directly [see below].
+ */
+static
+int
+d9chu(const double a, const double b, const double x, gsl_sf_result * result)
+{
+ const double EPS = 8.0 * GSL_DBL_EPSILON; /* EPS = 4.0D0*D1MACH(4) */
+ const int maxiter = 500;
+ double aa[4], bb[4];
+ int i;
+
+ double bp = 1.0 + a - b;
+ double ab = a*bp;
+ double ct2 = 2.0 * (x - ab);
+ double sab = a + bp;
+
+ double ct3 = sab + 1.0 + ab;
+ double anbn = ct3 + sab + 3.0;
+ double ct1 = 1.0 + 2.0*x/anbn;
+
+ bb[0] = 1.0;
+ aa[0] = 1.0;
+
+ bb[1] = 1.0 + 2.0*x/ct3;
+ aa[1] = 1.0 + ct2/ct3;
+
+ bb[2] = 1.0 + 6.0*ct1*x/ct3;
+ aa[2] = 1.0 + 6.0*ab/anbn + 3.0*ct1*ct2/ct3;
+
+ for(i=4; i<maxiter; i++) {
+ int j;
+ double c2;
+ double d1z;
+ double g1, g2, g3;
+ double x2i1 = 2*i - 3;
+ ct1 = x2i1/(x2i1-2.0);
+ anbn += x2i1 + sab;
+ ct2 = (x2i1 - 1.0)/anbn;
+ c2 = x2i1*ct2 - 1.0;
+ d1z = 2.0*x2i1*x/anbn;
+
+ ct3 = sab*ct2;
+ g1 = d1z + ct1*(c2+ct3);
+ g2 = d1z - c2;
+ g3 = ct1*(1.0 - ct3 - 2.0*ct2);
+
+ bb[3] = g1*bb[2] + g2*bb[1] + g3*bb[0];
+ aa[3] = g1*aa[2] + g2*aa[1] + g3*aa[0];
+
+ if(fabs(aa[3]*bb[0]-aa[0]*bb[3]) < EPS*fabs(bb[3]*bb[0])) break;
+
+ for(j=0; j<3; j++) {
+ aa[j] = aa[j+1];
+ bb[j] = bb[j+1];
+ }
+ }
+
+ result->val = aa[3]/bb[3];
+ result->err = 8.0 * GSL_DBL_EPSILON * fabs(result->val);
+
+ if(i == maxiter) {
+ GSL_ERROR ("error", GSL_EMAXITER);
+ }
+ else {
+ return GSL_SUCCESS;
+ }
+}
+
+
+/* Evaluate asymptotic for z^a U(a,b,z) ~ 2F0(a,1+a-b,-1/z)
+ * We check for termination of the 2F0 as a special case.
+ * Assumes x > 0.
+ * Also assumes a,b are not too large compared to x.
+ */
+static
+int
+hyperg_zaU_asymp(const double a, const double b, const double x, gsl_sf_result *result)
+{
+ const double ap = a;
+ const double bp = 1.0 + a - b;
+ const double rintap = floor(ap + 0.5);
+ const double rintbp = floor(bp + 0.5);
+ const int ap_neg_int = ( ap < 0.0 && fabs(ap - rintap) < INT_THRESHOLD );
+ const int bp_neg_int = ( bp < 0.0 && fabs(bp - rintbp) < INT_THRESHOLD );
+
+ if(ap_neg_int || bp_neg_int) {
+ /* Evaluate 2F0 polynomial.
+ */
+ double mxi = -1.0/x;
+ double nmax = -(int)(GSL_MIN(ap,bp) - 0.1);
+ double tn = 1.0;
+ double sum = 1.0;
+ double n = 1.0;
+ double sum_err = 0.0;
+ while(n <= nmax) {
+ double apn = (ap+n-1.0);
+ double bpn = (bp+n-1.0);
+ tn *= ((apn/n)*mxi)*bpn;
+ sum += tn;
+ sum_err += 2.0 * GSL_DBL_EPSILON * fabs(tn);
+ n += 1.0;
+ }
+ result->val = sum;
+ result->err = sum_err;
+ result->err += 2.0 * GSL_DBL_EPSILON * (fabs(nmax)+1.0) * fabs(sum);
+ return GSL_SUCCESS;
+ }
+ else {
+ return d9chu(a,b,x,result);
+ }
+}
+
+
+/* Evaluate finite sum which appears below.
+ */
+static
+int
+hyperg_U_finite_sum(int N, double a, double b, double x, double xeps,
+ gsl_sf_result * result)
+{
+ int i;
+ double sum_val;
+ double sum_err;
+
+ if(N <= 0) {
+ double t_val = 1.0;
+ double t_err = 0.0;
+ gsl_sf_result poch;
+ int stat_poch;
+
+ sum_val = 1.0;
+ sum_err = 0.0;
+ for(i=1; i<= -N; i++) {
+ const double xi1 = i - 1;
+ const double mult = (a+xi1)*x/((b+xi1)*(xi1+1.0));
+ t_val *= mult;
+ t_err += fabs(mult) * t_err + fabs(t_val) * 8.0 * 2.0 * GSL_DBL_EPSILON;
+ sum_val += t_val;
+ sum_err += t_err;
+ }
+
+ stat_poch = gsl_sf_poch_e(1.0+a-b, -a, &poch);
+
+ result->val = sum_val * poch.val;
+ result->err = fabs(sum_val) * poch.err + sum_err * fabs(poch.val);
+ result->err += fabs(poch.val) * (fabs(N) + 2.0) * GSL_DBL_EPSILON * fabs(sum_val);
+ result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
+ result->err *= 2.0; /* FIXME: fudge factor... why is the error estimate too small? */
+ return stat_poch;
+ }
+ else {
+ const int M = N - 2;
+ if(M < 0) {
+ result->val = 0.0;
+ result->err = 0.0;
+ return GSL_SUCCESS;
+ }
+ else {
+ gsl_sf_result gbm1;
+ gsl_sf_result gamr;
+ int stat_gbm1;
+ int stat_gamr;
+ double t_val = 1.0;
+ double t_err = 0.0;
+
+ sum_val = 1.0;
+ sum_err = 0.0;
+ for(i=1; i<=M; i++) {
+ const double mult = (a-b+i)*x/((1.0-b+i)*i);
+ t_val *= mult;
+ t_err += t_err * fabs(mult) + fabs(t_val) * 8.0 * 2.0 * GSL_DBL_EPSILON;
+ sum_val += t_val;
+ sum_err += t_err;
+ }
+
+ stat_gbm1 = gsl_sf_gamma_e(b-1.0, &gbm1);
+ stat_gamr = gsl_sf_gammainv_e(a, &gamr);
+
+ if(stat_gbm1 == GSL_SUCCESS) {
+ gsl_sf_result powx1N;
+ int stat_p = gsl_sf_pow_int_e(x, 1-N, &powx1N);
+ double pe_val = powx1N.val * xeps;
+ double pe_err = powx1N.err * fabs(xeps) + 2.0 * GSL_DBL_EPSILON * fabs(pe_val);
+ double coeff_val = gbm1.val * gamr.val * pe_val;
+ double coeff_err = gbm1.err * fabs(gamr.val * pe_val)
+ + gamr.err * fabs(gbm1.val * pe_val)
+ + fabs(gbm1.val * gamr.val) * pe_err
+ + 2.0 * GSL_DBL_EPSILON * fabs(coeff_val);
+
+ result->val = sum_val * coeff_val;
+ result->err = fabs(sum_val) * coeff_err + sum_err * fabs(coeff_val);
+ result->err += 2.0 * GSL_DBL_EPSILON * (M+2.0) * fabs(result->val);
+ result->err *= 2.0; /* FIXME: fudge factor... why is the error estimate too small? */
+ return stat_p;
+ }
+ else {
+ result->val = 0.0;
+ result->err = 0.0;
+ return stat_gbm1;
+ }
+ }
+ }
+}
+
+
+/* Based on SLATEC DCHU() [W. Fullerton]
+ * Assumes x > 0.
+ * This is just a series summation method, and
+ * it is not good for large a.
+ *
+ * I patched up the window for 1+a-b near zero. [GJ]
+ */
+static
+int
+hyperg_U_series(const double a, const double b, const double x, gsl_sf_result * result)
+{
+ const double EPS = 2.0 * GSL_DBL_EPSILON; /* EPS = D1MACH(3) */
+ const double SQRT_EPS = M_SQRT2 * GSL_SQRT_DBL_EPSILON;
+
+ if(fabs(1.0 + a - b) < SQRT_EPS) {
+ /* Original Comment: ALGORITHM IS BAD WHEN 1+A-B IS NEAR ZERO FOR SMALL X
+ */
+ /* We can however do the following:
+ * U(a,b,x) = U(a,a+1,x) when 1+a-b=0
+ * and U(a,a+1,x) = x^(-a).
+ */
+ double lnr = -a * log(x);
+ int stat_e = gsl_sf_exp_e(lnr, result);
+ result->err += 2.0 * SQRT_EPS * fabs(result->val);
+ return stat_e;
+ }
+ else {
+ double aintb = ( b < 0.0 ? ceil(b-0.5) : floor(b+0.5) );
+ double beps = b - aintb;
+ int N = aintb;
+
+ double lnx = log(x);
+ double xeps = exp(-beps*lnx);
+
+ /* Evaluate finite sum.
+ */
+ gsl_sf_result sum;
+ int stat_sum = hyperg_U_finite_sum(N, a, b, x, xeps, &sum);
+
+
+ /* Evaluate infinite sum. */
+
+ int istrt = ( N < 1 ? 1-N : 0 );
+ double xi = istrt;
+
+ gsl_sf_result gamr;
+ gsl_sf_result powx;
+ int stat_gamr = gsl_sf_gammainv_e(1.0+a-b, &gamr);
+ int stat_powx = gsl_sf_pow_int_e(x, istrt, &powx);
+ double sarg = beps*M_PI;
+ double sfact = ( sarg != 0.0 ? sarg/sin(sarg) : 1.0 );
+ double factor_val = sfact * ( GSL_IS_ODD(N) ? -1.0 : 1.0 ) * gamr.val * powx.val;
+ double factor_err = fabs(gamr.val) * powx.err + fabs(powx.val) * gamr.err
+ + 2.0 * GSL_DBL_EPSILON * fabs(factor_val);
+
+ gsl_sf_result pochai;
+ gsl_sf_result gamri1;
+ gsl_sf_result gamrni;
+ int stat_pochai = gsl_sf_poch_e(a, xi, &pochai);
+ int stat_gamri1 = gsl_sf_gammainv_e(xi + 1.0, &gamri1);
+ int stat_gamrni = gsl_sf_gammainv_e(aintb + xi, &gamrni);
+ int stat_gam123 = GSL_ERROR_SELECT_3(stat_gamr, stat_gamri1, stat_gamrni);
+ int stat_gamall = GSL_ERROR_SELECT_4(stat_sum, stat_gam123, stat_pochai, stat_powx);
+
+ gsl_sf_result pochaxibeps;
+ gsl_sf_result gamrxi1beps;
+ int stat_pochaxibeps = gsl_sf_poch_e(a, xi-beps, &pochaxibeps);
+ int stat_gamrxi1beps = gsl_sf_gammainv_e(xi + 1.0 - beps, &gamrxi1beps);
+
+ int stat_all = GSL_ERROR_SELECT_3(stat_gamall, stat_pochaxibeps, stat_gamrxi1beps);
+
+ double b0_val = factor_val * pochaxibeps.val * gamrni.val * gamrxi1beps.val;
+ double b0_err = fabs(factor_val * pochaxibeps.val * gamrni.val) * gamrxi1beps.err
+ + fabs(factor_val * pochaxibeps.val * gamrxi1beps.val) * gamrni.err
+ + fabs(factor_val * gamrni.val * gamrxi1beps.val) * pochaxibeps.err
+ + fabs(pochaxibeps.val * gamrni.val * gamrxi1beps.val) * factor_err
+ + 2.0 * GSL_DBL_EPSILON * fabs(b0_val);
+
+ if(fabs(xeps-1.0) < 0.5) {
+ /*
+ C X**(-BEPS) IS CLOSE TO 1.0D0, SO WE MUST BE
+ C CAREFUL IN EVALUATING THE DIFFERENCES.
+ */
+ int i;
+ gsl_sf_result pch1ai;
+ gsl_sf_result pch1i;
+ gsl_sf_result poch1bxibeps;
+ int stat_pch1ai = gsl_sf_pochrel_e(a + xi, -beps, &pch1ai);
+ int stat_pch1i = gsl_sf_pochrel_e(xi + 1.0 - beps, beps, &pch1i);
+ int stat_poch1bxibeps = gsl_sf_pochrel_e(b+xi, -beps, &poch1bxibeps);
+ double c0_t1_val = beps*pch1ai.val*pch1i.val;
+ double c0_t1_err = fabs(beps) * fabs(pch1ai.val) * pch1i.err
+ + fabs(beps) * fabs(pch1i.val) * pch1ai.err
+ + 2.0 * GSL_DBL_EPSILON * fabs(c0_t1_val);
+ double c0_t2_val = -poch1bxibeps.val + pch1ai.val - pch1i.val + c0_t1_val;
+ double c0_t2_err = poch1bxibeps.err + pch1ai.err + pch1i.err + c0_t1_err
+ + 2.0 * GSL_DBL_EPSILON * fabs(c0_t2_val);
+ double c0_val = factor_val * pochai.val * gamrni.val * gamri1.val * c0_t2_val;
+ double c0_err = fabs(factor_val * pochai.val * gamrni.val * gamri1.val) * c0_t2_err
+ + fabs(factor_val * pochai.val * gamrni.val * c0_t2_val) * gamri1.err
+ + fabs(factor_val * pochai.val * gamri1.val * c0_t2_val) * gamrni.err
+ + fabs(factor_val * gamrni.val * gamri1.val * c0_t2_val) * pochai.err
+ + fabs(pochai.val * gamrni.val * gamri1.val * c0_t2_val) * factor_err
+ + 2.0 * GSL_DBL_EPSILON * fabs(c0_val);
+ /*
+ C XEPS1 = (1.0 - X**(-BEPS))/BEPS = (X**(-BEPS) - 1.0)/(-BEPS)
+ */
+ gsl_sf_result dexprl;
+ int stat_dexprl = gsl_sf_exprel_e(-beps*lnx, &dexprl);
+ double xeps1_val = lnx * dexprl.val;
+ double xeps1_err = 2.0 * GSL_DBL_EPSILON * (1.0 + fabs(beps*lnx)) * fabs(dexprl.val)
+ + fabs(lnx) * dexprl.err
+ + 2.0 * GSL_DBL_EPSILON * fabs(xeps1_val);
+ double dchu_val = sum.val + c0_val + xeps1_val*b0_val;
+ double dchu_err = sum.err + c0_err
+ + fabs(xeps1_val)*b0_err + xeps1_err * fabs(b0_val)
+ + fabs(b0_val*lnx)*dexprl.err
+ + 2.0 * GSL_DBL_EPSILON * (fabs(sum.val) + fabs(c0_val) + fabs(xeps1_val*b0_val));
+ double xn = N;
+ double t_val;
+ double t_err;
+
+ stat_all = GSL_ERROR_SELECT_5(stat_all, stat_dexprl, stat_poch1bxibeps, stat_pch1i, stat_pch1ai);
+
+ for(i=1; i<2000; i++) {
+ const double xi = istrt + i;
+ const double xi1 = istrt + i - 1;
+ const double tmp = (a-1.0)*(xn+2.0*xi-1.0) + xi*(xi-beps);
+ const double b0_multiplier = (a+xi1-beps)*x/((xn+xi1)*(xi-beps));
+ const double c0_multiplier_1 = (a+xi1)*x/((b+xi1)*xi);
+ const double c0_multiplier_2 = tmp / (xi*(b+xi1)*(a+xi1-beps));
+ b0_val *= b0_multiplier;
+ b0_err += fabs(b0_multiplier) * b0_err + fabs(b0_val) * 8.0 * 2.0 * GSL_DBL_EPSILON;
+ c0_val = c0_multiplier_1 * c0_val - c0_multiplier_2 * b0_val;
+ c0_err = fabs(c0_multiplier_1) * c0_err
+ + fabs(c0_multiplier_2) * b0_err
+ + fabs(c0_val) * 8.0 * 2.0 * GSL_DBL_EPSILON
+ + fabs(b0_val * c0_multiplier_2) * 16.0 * 2.0 * GSL_DBL_EPSILON;
+ t_val = c0_val + xeps1_val*b0_val;
+ t_err = c0_err + fabs(xeps1_val)*b0_err;
+ t_err += fabs(b0_val*lnx) * dexprl.err;
+ t_err += fabs(b0_val)*xeps1_err;
+ dchu_val += t_val;
+ dchu_err += t_err;
+ if(fabs(t_val) < EPS*fabs(dchu_val)) break;
+ }
+
+ result->val = dchu_val;
+ result->err = 2.0 * dchu_err;
+ result->err += 2.0 * fabs(t_val);
+ result->err += 4.0 * GSL_DBL_EPSILON * (i+2.0) * fabs(dchu_val);
+ result->err *= 2.0; /* FIXME: fudge factor */
+
+ if(i >= 2000) {
+ GSL_ERROR ("error", GSL_EMAXITER);
+ }
+ else {
+ return stat_all;
+ }
+ }
+ else {
+ /*
+ C X**(-BEPS) IS VERY DIFFERENT FROM 1.0, SO THE
+ C STRAIGHTFORWARD FORMULATION IS STABLE.
+ */
+ int i;
+ double dchu_val;
+ double dchu_err;
+ double t_val;
+ double t_err;
+ gsl_sf_result dgamrbxi;
+ int stat_dgamrbxi = gsl_sf_gammainv_e(b+xi, &dgamrbxi);
+ double a0_val = factor_val * pochai.val * dgamrbxi.val * gamri1.val / beps;
+ double a0_err = fabs(factor_val * pochai.val * dgamrbxi.val / beps) * gamri1.err
+ + fabs(factor_val * pochai.val * gamri1.val / beps) * dgamrbxi.err
+ + fabs(factor_val * dgamrbxi.val * gamri1.val / beps) * pochai.err
+ + fabs(pochai.val * dgamrbxi.val * gamri1.val / beps) * factor_err
+ + 2.0 * GSL_DBL_EPSILON * fabs(a0_val);
+ stat_all = GSL_ERROR_SELECT_2(stat_all, stat_dgamrbxi);
+
+ b0_val = xeps * b0_val / beps;
+ b0_err = fabs(xeps / beps) * b0_err + 4.0 * GSL_DBL_EPSILON * fabs(b0_val);
+ dchu_val = sum.val + a0_val - b0_val;
+ dchu_err = sum.err + a0_err + b0_err
+ + 2.0 * GSL_DBL_EPSILON * (fabs(sum.val) + fabs(a0_val) + fabs(b0_val));
+
+ for(i=1; i<2000; i++) {
+ double xi = istrt + i;
+ double xi1 = istrt + i - 1;
+ double a0_multiplier = (a+xi1)*x/((b+xi1)*xi);
+ double b0_multiplier = (a+xi1-beps)*x/((aintb+xi1)*(xi-beps));
+ a0_val *= a0_multiplier;
+ a0_err += fabs(a0_multiplier) * a0_err;
+ b0_val *= b0_multiplier;
+ b0_err += fabs(b0_multiplier) * b0_err;
+ t_val = a0_val - b0_val;
+ t_err = a0_err + b0_err;
+ dchu_val += t_val;
+ dchu_err += t_err;
+ if(fabs(t_val) < EPS*fabs(dchu_val)) break;
+ }
+
+ result->val = dchu_val;
+ result->err = 2.0 * dchu_err;
+ result->err += 2.0 * fabs(t_val);
+ result->err += 4.0 * GSL_DBL_EPSILON * (i+2.0) * fabs(dchu_val);
+ result->err *= 2.0; /* FIXME: fudge factor */
+
+ if(i >= 2000) {
+ GSL_ERROR ("error", GSL_EMAXITER);
+ }
+ else {
+ return stat_all;
+ }
+ }
+ }
+}
+
+
+/* Assumes b > 0 and x > 0.
+ */
+static
+int
+hyperg_U_small_ab(const double a, const double b, const double x, gsl_sf_result * result)
+{
+ if(a == -1.0) {
+ /* U(-1,c+1,x) = Laguerre[c,0,x] = -b + x
+ */
+ result->val = -b + x;
+ result->err = 2.0 * GSL_DBL_EPSILON * (fabs(b) + fabs(x));
+ result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
+ return GSL_SUCCESS;
+ }
+ else if(a == 0.0) {
+ /* U(0,c+1,x) = Laguerre[c,0,x] = 1
+ */
+ result->val = 1.0;
+ result->err = 0.0;
+ return GSL_SUCCESS;
+ }
+ else if(ASYMP_EVAL_OK(a,b,x)) {
+ double p = pow(x, -a);
+ gsl_sf_result asymp;
+ int stat_asymp = hyperg_zaU_asymp(a, b, x, &asymp);
+ result->val = asymp.val * p;
+ result->err = asymp.err * p;
+ result->err += fabs(asymp.val) * GSL_DBL_EPSILON * fabs(a) * p;
+ result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
+ return stat_asymp;
+ }
+ else {
+ return hyperg_U_series(a, b, x, result);
+ }
+}
+
+
+/* Assumes b > 0 and x > 0.
+ */
+static
+int
+hyperg_U_small_a_bgt0(const double a, const double b, const double x,
+ gsl_sf_result * result,
+ double * ln_multiplier
+ )
+{
+ if(a == 0.0) {
+ result->val = 1.0;
+ result->err = 1.0;
+ *ln_multiplier = 0.0;
+ return GSL_SUCCESS;
+ }
+ else if( (b > 5000.0 && x < 0.90 * fabs(b))
+ || (b > 500.0 && x < 0.50 * fabs(b))
+ ) {
+ int stat = gsl_sf_hyperg_U_large_b_e(a, b, x, result, ln_multiplier);
+ if(stat == GSL_EOVRFLW)
+ return GSL_SUCCESS;
+ else
+ return stat;
+ }
+ else if(b > 15.0) {
+ /* Recurse up from b near 1.
+ */
+ double eps = b - floor(b);
+ double b0 = 1.0 + eps;
+ gsl_sf_result r_Ubm1;
+ gsl_sf_result r_Ub;
+ int stat_0 = hyperg_U_small_ab(a, b0, x, &r_Ubm1);
+ int stat_1 = hyperg_U_small_ab(a, b0+1.0, x, &r_Ub);
+ double Ubm1 = r_Ubm1.val;
+ double Ub = r_Ub.val;
+ double Ubp1;
+ double bp;
+
+ for(bp = b0+1.0; bp<b-0.1; bp += 1.0) {
+ Ubp1 = ((1.0+a-bp)*Ubm1 + (bp+x-1.0)*Ub)/x;
+ Ubm1 = Ub;
+ Ub = Ubp1;
+ }
+ result->val = Ub;
+ result->err = (fabs(r_Ubm1.err/r_Ubm1.val) + fabs(r_Ub.err/r_Ub.val)) * fabs(Ub);
+ result->err += 2.0 * GSL_DBL_EPSILON * (fabs(b-b0)+1.0) * fabs(Ub);
+ *ln_multiplier = 0.0;
+ return GSL_ERROR_SELECT_2(stat_0, stat_1);
+ }
+ else {
+ *ln_multiplier = 0.0;
+ return hyperg_U_small_ab(a, b, x, result);
+ }
+}
+
+
+/* We use this to keep track of large
+ * dynamic ranges in the recursions.
+ * This can be important because sometimes
+ * we want to calculate a very large and
+ * a very small number and the answer is
+ * the product, of order 1. This happens,
+ * for instance, when we apply a Kummer
+ * transform to make b positive and
+ * both x and b are large.
+ */
+#define RESCALE_2(u0,u1,factor,count) \
+do { \
+ double au0 = fabs(u0); \
+ if(au0 > factor) { \
+ u0 /= factor; \
+ u1 /= factor; \
+ count++; \
+ } \
+ else if(au0 < 1.0/factor) { \
+ u0 *= factor; \
+ u1 *= factor; \
+ count--; \
+ } \
+} while (0)
+
+
+/* Specialization to b >= 1, for integer parameters.
+ * Assumes x > 0.
+ */
+static
+int
+hyperg_U_int_bge1(const int a, const int b, const double x,
+ gsl_sf_result_e10 * result)
+{
+ if(a == 0) {
+ result->val = 1.0;
+ result->err = 0.0;
+ result->e10 = 0;
+ return GSL_SUCCESS;
+ }
+ else if(a == -1) {
+ result->val = -b + x;
+ result->err = 2.0 * GSL_DBL_EPSILON * (fabs(b) + fabs(x));
+ result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
+ result->e10 = 0;
+ return GSL_SUCCESS;
+ }
+ else if(b == a + 1) {
+ /* U(a,a+1,x) = x^(-a)
+ */
+ return gsl_sf_exp_e10_e(-a*log(x), result);
+ }
+ else if(ASYMP_EVAL_OK(a,b,x)) {
+ const double ln_pre_val = -a*log(x);
+ const double ln_pre_err = 2.0 * GSL_DBL_EPSILON * fabs(ln_pre_val);
+ gsl_sf_result asymp;
+ int stat_asymp = hyperg_zaU_asymp(a, b, x, &asymp);
+ int stat_e = gsl_sf_exp_mult_err_e10_e(ln_pre_val, ln_pre_err,
+ asymp.val, asymp.err,
+ result);
+ return GSL_ERROR_SELECT_2(stat_e, stat_asymp);
+ }
+ else if(SERIES_EVAL_OK(a,b,x)) {
+ gsl_sf_result ser;
+ const int stat_ser = hyperg_U_series(a, b, x, &ser);
+ result->val = ser.val;
+ result->err = ser.err;
+ result->e10 = 0;
+ return stat_ser;
+ }
+ else if(a < 0) {
+ /* Recurse backward from a = -1,0.
+ */
+ int scale_count = 0;
+ const double scale_factor = GSL_SQRT_DBL_MAX;
+ gsl_sf_result lnm;
+ gsl_sf_result y;
+ double lnscale;
+ double Uap1 = 1.0; /* U(0,b,x) */
+ double Ua = -b + x; /* U(-1,b,x) */
+ double Uam1;
+ int ap;
+
+ for(ap=-1; ap>a; ap--) {
+ Uam1 = ap*(b-ap-1.0)*Uap1 + (x+2.0*ap-b)*Ua;
+ Uap1 = Ua;
+ Ua = Uam1;
+ RESCALE_2(Ua,Uap1,scale_factor,scale_count);
+ }
+
+ lnscale = log(scale_factor);
+ lnm.val = scale_count*lnscale;
+ lnm.err = 2.0 * GSL_DBL_EPSILON * fabs(lnm.val);
+ y.val = Ua;
+ y.err = 4.0 * GSL_DBL_EPSILON * (fabs(a)+1.0) * fabs(Ua);
+ return gsl_sf_exp_mult_err_e10_e(lnm.val, lnm.err, y.val, y.err, result);
+ }
+ else if(b >= 2.0*a + x) {
+ /* Recurse forward from a = 0,1.
+ */
+ int scale_count = 0;
+ const double scale_factor = GSL_SQRT_DBL_MAX;
+ gsl_sf_result r_Ua;
+ gsl_sf_result lnm;
+ gsl_sf_result y;
+ double lnscale;
+ double lm;
+ int stat_1 = hyperg_U_small_a_bgt0(1.0, b, x, &r_Ua, &lm); /* U(1,b,x) */
+ int stat_e;
+ double Uam1 = 1.0; /* U(0,b,x) */
+ double Ua = r_Ua.val;
+ double Uap1;
+ int ap;
+
+ Uam1 *= exp(-lm);
+
+ for(ap=1; ap<a; ap++) {
+ Uap1 = -(Uam1 + (b-2.0*ap-x)*Ua)/(ap*(1.0+ap-b));
+ Uam1 = Ua;
+ Ua = Uap1;
+ RESCALE_2(Ua,Uam1,scale_factor,scale_count);
+ }
+
+ lnscale = log(scale_factor);
+ lnm.val = lm + scale_count * lnscale;
+ lnm.err = 2.0 * GSL_DBL_EPSILON * (fabs(lm) + fabs(scale_count*lnscale));
+ y.val = Ua;
+ y.err = fabs(r_Ua.err/r_Ua.val) * fabs(Ua);
+ y.err += 2.0 * GSL_DBL_EPSILON * (fabs(a) + 1.0) * fabs(Ua);
+ stat_e = gsl_sf_exp_mult_err_e10_e(lnm.val, lnm.err, y.val, y.err, result);
+ return GSL_ERROR_SELECT_2(stat_e, stat_1);
+ }
+ else {
+ if(b <= x) {
+ /* Recurse backward either to the b=a+1 line
+ * or to a=0, whichever we hit.
+ */
+ const double scale_factor = GSL_SQRT_DBL_MAX;
+ int scale_count = 0;
+ int stat_CF1;
+ double ru;
+ int CF1_count;
+ int a_target;
+ double lnU_target;
+ double Ua;
+ double Uap1;
+ double Uam1;
+ int ap;
+
+ if(b < a + 1) {
+ a_target = b-1;
+ lnU_target = -a_target*log(x);
+ }
+ else {
+ a_target = 0;
+ lnU_target = 0.0;
+ }
+
+ stat_CF1 = hyperg_U_CF1(a, b, 0, x, &ru, &CF1_count);
+
+ Ua = 1.0;
+ Uap1 = ru/a * Ua;
+ for(ap=a; ap>a_target; ap--) {
+ Uam1 = -((b-2.0*ap-x)*Ua + ap*(1.0+ap-b)*Uap1);
+ Uap1 = Ua;
+ Ua = Uam1;
+ RESCALE_2(Ua,Uap1,scale_factor,scale_count);
+ }
+
+ if(Ua == 0.0) {
+ result->val = 0.0;
+ result->err = 0.0;
+ result->e10 = 0;
+ GSL_ERROR ("error", GSL_EZERODIV);
+ }
+ else {
+ double lnscl = -scale_count*log(scale_factor);
+ double lnpre_val = lnU_target + lnscl;
+ double lnpre_err = 2.0 * GSL_DBL_EPSILON * (fabs(lnU_target) + fabs(lnscl));
+ double oUa_err = 2.0 * (fabs(a_target-a) + CF1_count + 1.0) * GSL_DBL_EPSILON * fabs(1.0/Ua);
+ int stat_e = gsl_sf_exp_mult_err_e10_e(lnpre_val, lnpre_err,
+ 1.0/Ua, oUa_err,
+ result);
+ return GSL_ERROR_SELECT_2(stat_e, stat_CF1);
+ }
+ }
+ else {
+ /* Recurse backward to near the b=2a+x line, then
+ * determine normalization by either direct evaluation
+ * or by a forward recursion. The direct evaluation
+ * is needed when x is small (which is precisely
+ * when it is easy to do).
+ */
+ const double scale_factor = GSL_SQRT_DBL_MAX;
+ int scale_count_for = 0;
+ int scale_count_bck = 0;
+ int a0 = 1;
+ int a1 = a0 + ceil(0.5*(b-x) - a0);
+ double Ua1_bck_val;
+ double Ua1_bck_err;
+ double Ua1_for_val;
+ double Ua1_for_err;
+ int stat_for;
+ int stat_bck;
+ gsl_sf_result lm_for;
+
+ {
+ /* Recurse back to determine U(a1,b), sans normalization.
+ */
+ double ru;
+ int CF1_count;
+ int stat_CF1 = hyperg_U_CF1(a, b, 0, x, &ru, &CF1_count);
+ double Ua = 1.0;
+ double Uap1 = ru/a * Ua;
+ double Uam1;
+ int ap;
+ for(ap=a; ap>a1; ap--) {
+ Uam1 = -((b-2.0*ap-x)*Ua + ap*(1.0+ap-b)*Uap1);
+ Uap1 = Ua;
+ Ua = Uam1;
+ RESCALE_2(Ua,Uap1,scale_factor,scale_count_bck);
+ }
+ Ua1_bck_val = Ua;
+ Ua1_bck_err = 2.0 * GSL_DBL_EPSILON * (fabs(a1-a)+CF1_count+1.0) * fabs(Ua);
+ stat_bck = stat_CF1;
+ }
+
+ if(b == 2*a1 && a1 > 1) {
+ /* This can happen when x is small, which is
+ * precisely when we need to be careful with
+ * this evaluation.
+ */
+ hyperg_lnU_beq2a((double)a1, x, &lm_for);
+ Ua1_for_val = 1.0;
+ Ua1_for_err = 0.0;
+ stat_for = GSL_SUCCESS;
+ }
+ else if(b == 2*a1 - 1 && a1 > 1) {
+ /* Similar to the above. Happens when x is small.
+ * Use
+ * U(a,2a-1) = (x U(a,2a) - U(a-1,2(a-1))) / (2a - 2)
+ */
+ gsl_sf_result lnU00, lnU12;
+ gsl_sf_result U00, U12;
+ hyperg_lnU_beq2a(a1-1.0, x, &lnU00);
+ hyperg_lnU_beq2a(a1, x, &lnU12);
+ if(lnU00.val > lnU12.val) {
+ lm_for.val = lnU00.val;
+ lm_for.err = lnU00.err;
+ U00.val = 1.0;
+ U00.err = 0.0;
+ gsl_sf_exp_err_e(lnU12.val - lm_for.val, lnU12.err + lm_for.err, &U12);
+ }
+ else {
+ lm_for.val = lnU12.val;
+ lm_for.err = lnU12.err;
+ U12.val = 1.0;
+ U12.err = 0.0;
+ gsl_sf_exp_err_e(lnU00.val - lm_for.val, lnU00.err + lm_for.err, &U00);
+ }
+ Ua1_for_val = (x * U12.val - U00.val) / (2.0*a1 - 2.0);
+ Ua1_for_err = (fabs(x)*U12.err + U00.err) / fabs(2.0*a1 - 2.0);
+ Ua1_for_err += 2.0 * GSL_DBL_EPSILON * fabs(Ua1_for_val);
+ stat_for = GSL_SUCCESS;
+ }
+ else {
+ /* Recurse forward to determine U(a1,b) with
+ * absolute normalization.
+ */
+ gsl_sf_result r_Ua;
+ double Uam1 = 1.0; /* U(a0-1,b,x) = U(0,b,x) */
+ double Ua;
+ double Uap1;
+ int ap;
+ double lm_for_local;
+ stat_for = hyperg_U_small_a_bgt0(a0, b, x, &r_Ua, &lm_for_local); /* U(1,b,x) */
+ Ua = r_Ua.val;
+ Uam1 *= exp(-lm_for_local);
+ lm_for.val = lm_for_local;
+ lm_for.err = 0.0;
+
+ for(ap=a0; ap<a1; ap++) {
+ Uap1 = -(Uam1 + (b-2.0*ap-x)*Ua)/(ap*(1.0+ap-b));
+ Uam1 = Ua;
+ Ua = Uap1;
+ RESCALE_2(Ua,Uam1,scale_factor,scale_count_for);
+ }
+ Ua1_for_val = Ua;
+ Ua1_for_err = fabs(Ua) * fabs(r_Ua.err/r_Ua.val);
+ Ua1_for_err += 2.0 * GSL_DBL_EPSILON * (fabs(a1-a0)+1.0) * fabs(Ua1_for_val);
+ }
+
+ /* Now do the matching to produce the final result.
+ */
+ if(Ua1_bck_val == 0.0) {
+ result->val = 0.0;
+ result->err = 0.0;
+ result->e10 = 0;
+ GSL_ERROR ("error", GSL_EZERODIV);
+ }
+ else if(Ua1_for_val == 0.0) {
+ /* Should never happen. */
+ UNDERFLOW_ERROR_E10(result);
+ }
+ else {
+ double lns = (scale_count_for - scale_count_bck)*log(scale_factor);
+ double ln_for_val = log(fabs(Ua1_for_val));
+ double ln_for_err = GSL_DBL_EPSILON + fabs(Ua1_for_err/Ua1_for_val);
+ double ln_bck_val = log(fabs(Ua1_bck_val));
+ double ln_bck_err = GSL_DBL_EPSILON + fabs(Ua1_bck_err/Ua1_bck_val);
+ double lnr_val = lm_for.val + ln_for_val - ln_bck_val + lns;
+ double lnr_err = lm_for.err + ln_for_err + ln_bck_err
+ + 2.0 * GSL_DBL_EPSILON * (fabs(lm_for.val) + fabs(ln_for_val) + fabs(ln_bck_val) + fabs(lns));
+ double sgn = GSL_SIGN(Ua1_for_val) * GSL_SIGN(Ua1_bck_val);
+ int stat_e = gsl_sf_exp_err_e10_e(lnr_val, lnr_err, result);
+ result->val *= sgn;
+ return GSL_ERROR_SELECT_3(stat_e, stat_bck, stat_for);
+ }
+ }
+ }
+}
+
+
+/* Handle b >= 1 for generic a,b values.
+ */
+static
+int
+hyperg_U_bge1(const double a, const double b, const double x,
+ gsl_sf_result_e10 * result)
+{
+ const double rinta = floor(a+0.5);
+ const int a_neg_integer = (a < 0.0 && fabs(a - rinta) < INT_THRESHOLD);
+
+ if(a == 0.0) {
+ result->val = 1.0;
+ result->err = 0.0;
+ result->e10 = 0;
+ return GSL_SUCCESS;
+ }
+ else if(a_neg_integer && fabs(rinta) < INT_MAX) {
+ /* U(-n,b,x) = (-1)^n n! Laguerre[n,b-1,x]
+ */
+ const int n = -(int)rinta;
+ const double sgn = (GSL_IS_ODD(n) ? -1.0 : 1.0);
+ gsl_sf_result lnfact;
+ gsl_sf_result L;
+ const int stat_L = gsl_sf_laguerre_n_e(n, b-1.0, x, &L);
+ gsl_sf_lnfact_e(n, &lnfact);
+ {
+ const int stat_e = gsl_sf_exp_mult_err_e10_e(lnfact.val, lnfact.err,
+ sgn*L.val, L.err,
+ result);
+ return GSL_ERROR_SELECT_2(stat_e, stat_L);
+ }
+ }
+ else if(ASYMP_EVAL_OK(a,b,x)) {
+ const double ln_pre_val = -a*log(x);
+ const double ln_pre_err = 2.0 * GSL_DBL_EPSILON * fabs(ln_pre_val);
+ gsl_sf_result asymp;
+ int stat_asymp = hyperg_zaU_asymp(a, b, x, &asymp);
+ int stat_e = gsl_sf_exp_mult_err_e10_e(ln_pre_val, ln_pre_err,
+ asymp.val, asymp.err,
+ result);
+ return GSL_ERROR_SELECT_2(stat_e, stat_asymp);
+ }
+ else if(fabs(a) <= 1.0) {
+ gsl_sf_result rU;
+ double ln_multiplier;
+ int stat_U = hyperg_U_small_a_bgt0(a, b, x, &rU, &ln_multiplier);
+ int stat_e = gsl_sf_exp_mult_err_e10_e(ln_multiplier, 2.0*GSL_DBL_EPSILON*fabs(ln_multiplier), rU.val, rU.err, result);
+ return GSL_ERROR_SELECT_2(stat_U, stat_e);
+ }
+ else if(SERIES_EVAL_OK(a,b,x)) {
+ gsl_sf_result ser;
+ const int stat_ser = hyperg_U_series(a, b, x, &ser);
+ result->val = ser.val;
+ result->err = ser.err;
+ result->e10 = 0;
+ return stat_ser;
+ }
+ else if(a < 0.0) {
+ /* Recurse backward on a and then upward on b.
+ */
+ const double scale_factor = GSL_SQRT_DBL_MAX;
+ const double a0 = a - floor(a) - 1.0;
+ const double b0 = b - floor(b) + 1.0;
+ int scale_count = 0;
+ double lm_0, lm_1;
+ double lm_max;
+ gsl_sf_result r_Uap1;
+ gsl_sf_result r_Ua;
+ int stat_0 = hyperg_U_small_a_bgt0(a0+1.0, b0, x, &r_Uap1, &lm_0);
+ int stat_1 = hyperg_U_small_a_bgt0(a0, b0, x, &r_Ua, &lm_1);
+ int stat_e;
+ double Uap1 = r_Uap1.val;
+ double Ua = r_Ua.val;
+ double Uam1;
+ double ap;
+ lm_max = GSL_MAX(lm_0, lm_1);
+ Uap1 *= exp(lm_0-lm_max);
+ Ua *= exp(lm_1-lm_max);
+
+ /* Downward recursion on a.
+ */
+ for(ap=a0; ap>a+0.1; ap -= 1.0) {
+ Uam1 = ap*(b0-ap-1.0)*Uap1 + (x+2.0*ap-b0)*Ua;
+ Uap1 = Ua;
+ Ua = Uam1;
+ RESCALE_2(Ua,Uap1,scale_factor,scale_count);
+ }
+
+ if(b < 2.0) {
+ /* b == b0, so no recursion necessary
+ */
+ const double lnscale = log(scale_factor);
+ gsl_sf_result lnm;
+ gsl_sf_result y;
+ lnm.val = lm_max + scale_count * lnscale;
+ lnm.err = 2.0 * GSL_DBL_EPSILON * (fabs(lm_max) + scale_count * fabs(lnscale));
+ y.val = Ua;
+ y.err = fabs(r_Uap1.err/r_Uap1.val) * fabs(Ua);
+ y.err += fabs(r_Ua.err/r_Ua.val) * fabs(Ua);
+ y.err += 2.0 * GSL_DBL_EPSILON * (fabs(a-a0) + 1.0) * fabs(Ua);
+ y.err *= fabs(lm_0-lm_max) + 1.0;
+ y.err *= fabs(lm_1-lm_max) + 1.0;
+ stat_e = gsl_sf_exp_mult_err_e10_e(lnm.val, lnm.err, y.val, y.err, result);
+ }
+ else {
+ /* Upward recursion on b.
+ */
+ const double err_mult = fabs(b-b0) + fabs(a-a0) + 1.0;
+ const double lnscale = log(scale_factor);
+ gsl_sf_result lnm;
+ gsl_sf_result y;
+
+ double Ubm1 = Ua; /* U(a,b0) */
+ double Ub = (a*(b0-a-1.0)*Uap1 + (a+x)*Ua)/x; /* U(a,b0+1) */
+ double Ubp1;
+ double bp;
+ for(bp=b0+1.0; bp<b-0.1; bp += 1.0) {
+ Ubp1 = ((1.0+a-bp)*Ubm1 + (bp+x-1.0)*Ub)/x;
+ Ubm1 = Ub;
+ Ub = Ubp1;
+ RESCALE_2(Ub,Ubm1,scale_factor,scale_count);
+ }
+
+ lnm.val = lm_max + scale_count * lnscale;
+ lnm.err = 2.0 * GSL_DBL_EPSILON * (fabs(lm_max) + fabs(scale_count * lnscale));
+ y.val = Ub;
+ y.err = 2.0 * err_mult * fabs(r_Uap1.err/r_Uap1.val) * fabs(Ub);
+ y.err += 2.0 * err_mult * fabs(r_Ua.err/r_Ua.val) * fabs(Ub);
+ y.err += 2.0 * GSL_DBL_EPSILON * err_mult * fabs(Ub);
+ y.err *= fabs(lm_0-lm_max) + 1.0;
+ y.err *= fabs(lm_1-lm_max) + 1.0;
+ stat_e = gsl_sf_exp_mult_err_e10_e(lnm.val, lnm.err, y.val, y.err, result);
+ }
+ return GSL_ERROR_SELECT_3(stat_e, stat_0, stat_1);
+ }
+ else if(b >= 2*a + x) {
+ /* Recurse forward from a near zero.
+ * Note that we cannot cross the singularity at
+ * the line b=a+1, because the only way we could
+ * be in that little wedge is if a < 1. But we
+ * have already dealt with the small a case.
+ */
+ int scale_count = 0;
+ const double a0 = a - floor(a);
+ const double scale_factor = GSL_SQRT_DBL_MAX;
+ double lnscale;
+ double lm_0, lm_1, lm_max;
+ gsl_sf_result r_Uam1;
+ gsl_sf_result r_Ua;
+ int stat_0 = hyperg_U_small_a_bgt0(a0-1.0, b, x, &r_Uam1, &lm_0);
+ int stat_1 = hyperg_U_small_a_bgt0(a0, b, x, &r_Ua, &lm_1);
+ int stat_e;
+ gsl_sf_result lnm;
+ gsl_sf_result y;
+ double Uam1 = r_Uam1.val;
+ double Ua = r_Ua.val;
+ double Uap1;
+ double ap;
+ lm_max = GSL_MAX(lm_0, lm_1);
+ Uam1 *= exp(lm_0-lm_max);
+ Ua *= exp(lm_1-lm_max);
+
+ for(ap=a0; ap<a-0.1; ap += 1.0) {
+ Uap1 = -(Uam1 + (b-2.0*ap-x)*Ua)/(ap*(1.0+ap-b));
+ Uam1 = Ua;
+ Ua = Uap1;
+ RESCALE_2(Ua,Uam1,scale_factor,scale_count);
+ }
+
+ lnscale = log(scale_factor);
+ lnm.val = lm_max + scale_count * lnscale;
+ lnm.err = 2.0 * GSL_DBL_EPSILON * (fabs(lm_max) + fabs(scale_count * lnscale));
+ y.val = Ua;
+ y.err = fabs(r_Uam1.err/r_Uam1.val) * fabs(Ua);
+ y.err += fabs(r_Ua.err/r_Ua.val) * fabs(Ua);
+ y.err += 2.0 * GSL_DBL_EPSILON * (fabs(a-a0) + 1.0) * fabs(Ua);
+ y.err *= fabs(lm_0-lm_max) + 1.0;
+ y.err *= fabs(lm_1-lm_max) + 1.0;
+ stat_e = gsl_sf_exp_mult_err_e10_e(lnm.val, lnm.err, y.val, y.err, result);
+ return GSL_ERROR_SELECT_3(stat_e, stat_0, stat_1);
+ }
+ else {
+ if(b <= x) {
+ /* Recurse backward to a near zero.
+ */
+ const double a0 = a - floor(a);
+ const double scale_factor = GSL_SQRT_DBL_MAX;
+ int scale_count = 0;
+ gsl_sf_result lnm;
+ gsl_sf_result y;
+ double lnscale;
+ double lm_0;
+ double Uap1;
+ double Ua;
+ double Uam1;
+ gsl_sf_result U0;
+ double ap;
+ double ru;
+ double r;
+ int CF1_count;
+ int stat_CF1 = hyperg_U_CF1(a, b, 0, x, &ru, &CF1_count);
+ int stat_U0;
+ int stat_e;
+ r = ru/a;
+ Ua = GSL_SQRT_DBL_MIN;
+ Uap1 = r * Ua;
+ for(ap=a; ap>a0+0.1; ap -= 1.0) {
+ Uam1 = -((b-2.0*ap-x)*Ua + ap*(1.0+ap-b)*Uap1);
+ Uap1 = Ua;
+ Ua = Uam1;
+ RESCALE_2(Ua,Uap1,scale_factor,scale_count);
+ }
+
+ stat_U0 = hyperg_U_small_a_bgt0(a0, b, x, &U0, &lm_0);
+
+ lnscale = log(scale_factor);
+ lnm.val = lm_0 - scale_count * lnscale;
+ lnm.err = 2.0 * GSL_DBL_EPSILON * (fabs(lm_0) + fabs(scale_count * lnscale));
+ y.val = GSL_SQRT_DBL_MIN*(U0.val/Ua);
+ y.err = GSL_SQRT_DBL_MIN*(U0.err/fabs(Ua));
+ y.err += 2.0 * GSL_DBL_EPSILON * (fabs(a0-a) + CF1_count + 1.0) * fabs(y.val);
+ stat_e = gsl_sf_exp_mult_err_e10_e(lnm.val, lnm.err, y.val, y.err, result);
+ return GSL_ERROR_SELECT_3(stat_e, stat_U0, stat_CF1);
+ }
+ else {
+ /* Recurse backward to near the b=2a+x line, then
+ * forward from a near zero to get the normalization.
+ */
+ int scale_count_for = 0;
+ int scale_count_bck = 0;
+ const double scale_factor = GSL_SQRT_DBL_MAX;
+ const double eps = a - floor(a);
+ const double a0 = ( eps == 0.0 ? 1.0 : eps );
+ const double a1 = a0 + ceil(0.5*(b-x) - a0);
+ gsl_sf_result lnm;
+ gsl_sf_result y;
+ double lm_for;
+ double lnscale;
+ double Ua1_bck;
+ double Ua1_for;
+ int stat_for;
+ int stat_bck;
+ int stat_e;
+ int CF1_count;
+
+ {
+ /* Recurse back to determine U(a1,b), sans normalization.
+ */
+ double Uap1;
+ double Ua;
+ double Uam1;
+ double ap;
+ double ru;
+ double r;
+ int stat_CF1 = hyperg_U_CF1(a, b, 0, x, &ru, &CF1_count);
+ r = ru/a;
+ Ua = GSL_SQRT_DBL_MIN;
+ Uap1 = r * Ua;
+ for(ap=a; ap>a1+0.1; ap -= 1.0) {
+ Uam1 = -((b-2.0*ap-x)*Ua + ap*(1.0+ap-b)*Uap1);
+ Uap1 = Ua;
+ Ua = Uam1;
+ RESCALE_2(Ua,Uap1,scale_factor,scale_count_bck);
+ }
+ Ua1_bck = Ua;
+ stat_bck = stat_CF1;
+ }
+ {
+ /* Recurse forward to determine U(a1,b) with
+ * absolute normalization.
+ */
+ gsl_sf_result r_Uam1;
+ gsl_sf_result r_Ua;
+ double lm_0, lm_1;
+ int stat_0 = hyperg_U_small_a_bgt0(a0-1.0, b, x, &r_Uam1, &lm_0);
+ int stat_1 = hyperg_U_small_a_bgt0(a0, b, x, &r_Ua, &lm_1);
+ double Uam1 = r_Uam1.val;
+ double Ua = r_Ua.val;
+ double Uap1;
+ double ap;
+
+ lm_for = GSL_MAX(lm_0, lm_1);
+ Uam1 *= exp(lm_0 - lm_for);
+ Ua *= exp(lm_1 - lm_for);
+
+ for(ap=a0; ap<a1-0.1; ap += 1.0) {
+ Uap1 = -(Uam1 + (b-2.0*ap-x)*Ua)/(ap*(1.0+ap-b));
+ Uam1 = Ua;
+ Ua = Uap1;
+ RESCALE_2(Ua,Uam1,scale_factor,scale_count_for);
+ }
+ Ua1_for = Ua;
+ stat_for = GSL_ERROR_SELECT_2(stat_0, stat_1);
+ }
+
+ lnscale = log(scale_factor);
+ lnm.val = lm_for + (scale_count_for - scale_count_bck)*lnscale;
+ lnm.err = 2.0 * GSL_DBL_EPSILON * (fabs(lm_for) + fabs(scale_count_for - scale_count_bck)*fabs(lnscale));
+ y.val = GSL_SQRT_DBL_MIN*Ua1_for/Ua1_bck;
+ y.err = 2.0 * GSL_DBL_EPSILON * (fabs(a-a0) + CF1_count + 1.0) * fabs(y.val);
+ stat_e = gsl_sf_exp_mult_err_e10_e(lnm.val, lnm.err, y.val, y.err, result);
+ return GSL_ERROR_SELECT_3(stat_e, stat_bck, stat_for);
+ }
+ }
+}
+
+
+/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
+
+
+int
+gsl_sf_hyperg_U_int_e10_e(const int a, const int b, const double x,
+ gsl_sf_result_e10 * result)
+{
+ /* CHECK_POINTER(result) */
+
+ if(x <= 0.0) {
+ DOMAIN_ERROR_E10(result);
+ }
+ else {
+ if(b >= 1) {
+ return hyperg_U_int_bge1(a, b, x, result);
+ }
+ else {
+ /* Use the reflection formula
+ * U(a,b,x) = x^(1-b) U(1+a-b,2-b,x)
+ */
+ gsl_sf_result_e10 U;
+ double ln_x = log(x);
+ int ap = 1 + a - b;
+ int bp = 2 - b;
+ int stat_e;
+ int stat_U = hyperg_U_int_bge1(ap, bp, x, &U);
+ double ln_pre_val = (1.0-b)*ln_x;
+ double ln_pre_err = 2.0 * GSL_DBL_EPSILON * (fabs(b)+1.0) * fabs(ln_x);
+ ln_pre_err += 2.0 * GSL_DBL_EPSILON * fabs(1.0-b); /* error in log(x) */
+ stat_e = gsl_sf_exp_mult_err_e10_e(ln_pre_val + U.e10*M_LN10, ln_pre_err,
+ U.val, U.err,
+ result);
+ return GSL_ERROR_SELECT_2(stat_e, stat_U);
+ }
+ }
+}
+
+
+int
+gsl_sf_hyperg_U_e10_e(const double a, const double b, const double x,
+ gsl_sf_result_e10 * result)
+{
+ const double rinta = floor(a + 0.5);
+ const double rintb = floor(b + 0.5);
+ const int a_integer = ( fabs(a - rinta) < INT_THRESHOLD );
+ const int b_integer = ( fabs(b - rintb) < INT_THRESHOLD );
+
+ /* CHECK_POINTER(result) */
+
+ if(x <= 0.0) {
+ DOMAIN_ERROR_E10(result);
+ }
+ else if(a == 0.0) {
+ result->val = 1.0;
+ result->err = 0.0;
+ result->e10 = 0;
+ return GSL_SUCCESS;
+ }
+ else if(a_integer && b_integer) {
+ return gsl_sf_hyperg_U_int_e10_e(rinta, rintb, x, result);
+ }
+ else {
+ if(b >= 1.0) {
+ /* Use b >= 1 function.
+ */
+ return hyperg_U_bge1(a, b, x, result);
+ }
+ else {
+ /* Use the reflection formula
+ * U(a,b,x) = x^(1-b) U(1+a-b,2-b,x)
+ */
+ const double lnx = log(x);
+ const double ln_pre_val = (1.0-b)*lnx;
+ const double ln_pre_err = fabs(lnx) * 2.0 * GSL_DBL_EPSILON * (1.0 + fabs(b));
+ const double ap = 1.0 + a - b;
+ const double bp = 2.0 - b;
+ gsl_sf_result_e10 U;
+ int stat_U = hyperg_U_bge1(ap, bp, x, &U);
+ int stat_e = gsl_sf_exp_mult_err_e10_e(ln_pre_val + U.e10*M_LN10, ln_pre_err,
+ U.val, U.err,
+ result);
+ return GSL_ERROR_SELECT_2(stat_e, stat_U);
+ }
+ }
+}
+
+
+int
+gsl_sf_hyperg_U_int_e(const int a, const int b, const double x, gsl_sf_result * result)
+{
+ gsl_sf_result_e10 re;
+ int stat_U = gsl_sf_hyperg_U_int_e10_e(a, b, x, &re);
+ int stat_c = gsl_sf_result_smash_e(&re, result);
+ return GSL_ERROR_SELECT_2(stat_c, stat_U);
+}
+
+
+int
+gsl_sf_hyperg_U_e(const double a, const double b, const double x, gsl_sf_result * result)
+{
+ gsl_sf_result_e10 re;
+ int stat_U = gsl_sf_hyperg_U_e10_e(a, b, x, &re);
+ int stat_c = gsl_sf_result_smash_e(&re, result);
+ return GSL_ERROR_SELECT_2(stat_c, stat_U);
+}
+
+
+/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
+
+#include "eval.h"
+
+double gsl_sf_hyperg_U_int(const int a, const int b, const double x)
+{
+ EVAL_RESULT(gsl_sf_hyperg_U_int_e(a, b, x, &result));
+}
+
+double gsl_sf_hyperg_U(const double a, const double b, const double x)
+{
+ EVAL_RESULT(gsl_sf_hyperg_U_e(a, b, x, &result));
+}
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