/* specfunc/hyperg_U.c * * Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman * Copyright (C) 2009, 2010 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 #include #include #include #include #include #include #include #include #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; ival = 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; } } } } /* Evaluate infinite sum which appears below. */ static int hyperg_U_infinite_sum_stable(int N, double a, double bint, double b, double beps, double x, double xeps, gsl_sf_result sum, gsl_sf_result * result) { const double EPS = 2.0 * GSL_DBL_EPSILON; /* EPS = D1MACH(3) */ 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(bint + xi, &gamrni); int stat_gam123 = GSL_ERROR_SELECT_3(stat_gamr, stat_gamri1, stat_gamrni); int stat_gamall = GSL_ERROR_SELECT_3(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); /* 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/((bint+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; } } static int hyperg_U_infinite_sum_simple(int N, double a, double bint, double b, double beps, double x, double xeps, gsl_sf_result sum, gsl_sf_result * result) { const double EPS = 2.0 * GSL_DBL_EPSILON; /* EPS = D1MACH(3) */ int istrt = ( N < 1 ? 1-N : 0 ); double xi = istrt; gsl_sf_result powx; 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 ) * powx.val; double factor_err = fabs(powx.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(bint + xi, &gamrni); int stat_gam123 = GSL_ERROR_SELECT_2(stat_gamri1, stat_gamrni); int stat_gamall = GSL_ERROR_SELECT_3(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 X = sfact * ( GSL_IS_ODD(N) ? -1.0 : 1.0 ) * powx.val * gsl_sf_poch(1 + a - b, xi - 1 + b - beps) * gsl_sf_gammainv(a); double b0_val = X * 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); /* 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 gamr; gsl_sf_result dgamrbxi; int stat_gamr = gsl_sf_gammainv_e(1.0+a-b, &gamr); int stat_dgamrbxi = gsl_sf_gammainv_e(b+xi, &dgamrbxi); double a0_val = factor_val * gamr.val * pochai.val * dgamrbxi.val * gamri1.val / beps; double a0_err = fabs(factor_val * pochai.val * dgamrbxi.val * gamri1.val / beps) * gamr.err + fabs(factor_val * gamr.val * dgamrbxi.val * gamri1.val / beps) * pochai.err + fabs(factor_val * gamr.val * pochai.val * gamri1.val / beps) * dgamrbxi.err + fabs(factor_val * gamr.val * pochai.val * dgamrbxi.val / beps) * gamri1.err + fabs(pochai.val * gamr.val * dgamrbxi.val * gamri1.val / beps) * factor_err + 2.0 * GSL_DBL_EPSILON * fabs(a0_val); stat_all = GSL_ERROR_SELECT_3(stat_all, stat_gamr, 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/((bint+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(!gsl_finite(t_val) || 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; } } static int hyperg_U_infinite_sum_improved(int N, double a, double bint, double b, double beps, double x, double xeps, gsl_sf_result sum, gsl_sf_result * result) { const double EPS = 2.0 * GSL_DBL_EPSILON; /* EPS = D1MACH(3) */ const double lnx = log(x); 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(bint + xi, &gamrni); int stat_gam123 = GSL_ERROR_SELECT_3(stat_gamr, stat_gamri1, stat_gamrni); int stat_gamall = GSL_ERROR_SELECT_3(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); /* 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; } } /* 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 SQRT_EPS = M_SQRT2 * GSL_SQRT_DBL_EPSILON; double bint = ( b < 0.0 ? ceil(b-0.5) : floor(b+0.5) ); double beps = b - bint; double a_beps = a - beps; double r_a_beps = floor(a_beps + 0.5); double a_beps_int = ( fabs(a_beps - r_a_beps) < INT_THRESHOLD ); /* double a_b_1 = a-b+1; double r_a_b_1 = floor(a_b_1+0.5); double r_a_b_1_int = (fabs(a_b_1-r_a_b_1)< INT_THRESHOLD); Check for (a-beps) being a member of -N; N being 0,1,... */ if (a_beps_int && a_beps <= 0) { beps=beps - 1 + floor(a_beps);bint=bint + 1 - floor(a_beps); } 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 { int N = (int) bint; 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); int stat_inf; /* Evaluate infinite sum. */ if(fabs(xeps-1.0) > 0.5 ) { stat_inf = hyperg_U_infinite_sum_stable(N, a, bint, b, beps, x, xeps, sum, result); } else if (1+a-b < 0 && 1+a-b==floor(1+a-b) && beps != 0) { stat_inf = hyperg_U_infinite_sum_simple(N, a, bint, b, beps, x, xeps, sum, result); } else { stat_inf = hyperg_U_infinite_sum_improved(N, a, bint, b, beps, x, xeps, sum, result); } return GSL_ERROR_SELECT_2(stat_sum, stat_inf); } } /* 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 = 0.0; *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; bpval = 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) && 1 + a - b > 0) { 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; apa_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; apval = 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= 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; apa0+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; apval = factor * r1.val * r2.val; result->err = fabs(factor) * (r1.err + r2.err); result->e10 = 0; return GSL_ERROR_SELECT_2(stat_1, stat_2); } static int hyperg_U_int_origin (const int a, const int b, gsl_sf_result_e10 * result) { return hyperg_U_origin (a, b, result); } /* Calculate U(a,b,x) for x < 0 Abramowitz and Stegun formula 13.1.3 U(a,b,x) = (gamma(1-b)/gamma(1+a-b)) M(a,b,x) - z^(1-b) (gamma(1-b)/gamma(a)) M(1+a-b,2-b,x) can be transformed into U(a,b,x) = poch(1+a-b,-a) M(a,b,x) + z^(1-b) poch(a,-(1+a-b)) M(1+a-b,2-b,x) using the reflection formula 6.1.17 and the definition of Poch(a,b)=gamma(a+b)/gamma(a). Our poch function already handles the special cases of ratios of gamma functions with negative integer argument. Note that U(a,b,x) is complex in general for x<0 due to the term x^(1-b), but is real when 1) b is an integer 4) a is zero or a negative integer so x^(1-b)/gamma(a) is zero. For integer b U(a,b,x) is defined as the limit beta->b U(a,beta,x). This makes the situation slightly more complicated. */ static int hyperg_U_negx (const double a, const double b, const double x, gsl_sf_result_e10 * result) { gsl_sf_result r1, r2; int stat_1, stat_2, status; int a_int = (a == floor(a)); int b_int = (b == floor(b)); double T1 = 0, T1_err = 0, T2 = 0, T2_err = 0; /* Compute the first term poch(1+a-b) M(a,b,x) */ if (b_int && b <= 0 && !(a_int && a <= 0 && a >= b)) { /* Need to handle first term as lim_{beta->b} poch(1+a-beta,-a) M(a,beta,x) due to pole in M(a,b,x) for b == 0 or -ve integer We skip this case when a is zero or a negative integer and a>=b because the hypergeometric series terminates before any singular terms */ /* FIXME: TO BE IMPLEMENTED ! */ result->val = GSL_NAN; result->err = GSL_NAN; GSL_ERROR("limit case integer b <= 0 unimplemented", GSL_EUNIMPL); } else { stat_1 = gsl_sf_poch_e(1+a-b,-a,&r1); status = stat_1; if (r1.val != 0.0) { gsl_sf_result Mr1; int stat_Mr1 = gsl_sf_hyperg_1F1_e (a, b, x, &Mr1); status = GSL_ERROR_SELECT_2(status, stat_Mr1); T1 = Mr1.val * r1.val; T1_err = 2.0 * GSL_DBL_EPSILON * fabs(T1) + fabs(Mr1.err * r1.val) + fabs(Mr1.val * r1.err) ; } } /* Compute the second term z^(1-b) poch(a,-(1+a-b)) M(1+a-b,2-b,x) */ if (b_int && b >= 2 && !(a_int && a <= (b - 2))) { /* Need to handle second term as a limit due to pole in M(1+a-b,2-b,x). We skip this case when a is integer and a <= b-2 because the hypergeometric series terminates before any singular terms */ /* FIXME: TO BE IMPLEMENTED ! */ result->val = GSL_NAN; result->err = GSL_NAN; GSL_ERROR("limit case integer b >= 2 unimplemented", GSL_EUNIMPL); } else { if (a_int && a <= 0 && (b >= 1)) { r2.val = 0; r2.err = 0; } else { stat_2 = gsl_sf_poch_e(a,-(1+a-b),&r2); status = GSL_ERROR_SELECT_2(status, stat_2); } if (r2.val != 0.0) { gsl_sf_result Mr2; int stat_Mr2 = gsl_sf_hyperg_1F1_e (1+a-b, 2-b, x, &Mr2); T2 = Mr2.val * r2.val; T2_err = 2.0 * GSL_DBL_EPSILON * fabs(T2) + fabs(Mr2.err * r2.val) + fabs(Mr2.val * r2.err); status = GSL_ERROR_SELECT_2(status, stat_Mr2); if (T2 != 0.0) { double x1mb = pow(x, 1-b); T2 = x1mb * T2; T2_err = fabs(x1mb) * T2_err; } } } result->val = (T1 + T2); result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val) + (T1_err + T2_err); result->e10 = 0; return status; } static int hyperg_U_int_negx (const int a, const int b, const double x, gsl_sf_result_e10 * result) { /* Looking at the tests it seems that everything is handled correctly by hyperg_U_negx except aval = res_tem; result->err = res_tem_err; return status; } else { return hyperg_U_negx (a, b, x, result); } } /*-*-*-*-*-*-*-*-*-*-*-* 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 && b >= 1) { DOMAIN_ERROR_E10(result); } else if (x == 0.0) { return hyperg_U_int_origin (a, b, result); } else if (x < 0.0) { return hyperg_U_int_negx (a, b, x, 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 && b >= 1) { 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 (x == 0.0) { return hyperg_U_origin (a, b, result); } else if(a_integer && b == a + 1) /* This is DLMF 13.6.4 */ { gsl_sf_result powx1N_1; gsl_sf_pow_int_e(x, -a, &powx1N_1); result->val = powx1N_1.val; result->err = powx1N_1.err; 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 (x < 0.0) { return hyperg_U_negx (a, b, 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 = {0, 0, 0}; 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 = {0, 0, 0}; 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)); }