| /* specfunc/hyperg.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 */ |
| |
| /* Miscellaneous implementations of use |
| * for evaluation of hypergeometric functions. |
| */ |
| #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 "error.h" |
| #include "hyperg.h" |
| |
| #define SUM_LARGE (1.0e-5*GSL_DBL_MAX) |
| |
| |
| int |
| gsl_sf_hyperg_1F1_series_e(const double a, const double b, const double x, |
| gsl_sf_result * result |
| ) |
| { |
| double an = a; |
| double bn = b; |
| double n = 1.0; |
| double del = 1.0; |
| double abs_del = 1.0; |
| double max_abs_del = 1.0; |
| double sum_val = 1.0; |
| double sum_err = 0.0; |
| |
| while(abs_del/fabs(sum_val) > 0.25*GSL_DBL_EPSILON) { |
| double u, abs_u; |
| |
| if(bn == 0.0) { |
| DOMAIN_ERROR(result); |
| } |
| |
| if(an == 0.0) { |
| result->val = sum_val; |
| result->err = sum_err; |
| result->err += 2.0 * GSL_DBL_EPSILON * n * fabs(sum_val); |
| return GSL_SUCCESS; |
| } |
| |
| if (n > 10000.0) { |
| result->val = sum_val; |
| result->err = sum_err; |
| GSL_ERROR ("hypergeometric series failed to converge", GSL_EFAILED); |
| } |
| |
| u = x * (an/(bn*n)); |
| abs_u = fabs(u); |
| if(abs_u > 1.0 && max_abs_del > GSL_DBL_MAX/abs_u) { |
| result->val = sum_val; |
| result->err = fabs(sum_val); |
| GSL_ERROR ("overflow", GSL_EOVRFLW); |
| } |
| del *= u; |
| sum_val += del; |
| if(fabs(sum_val) > SUM_LARGE) { |
| result->val = sum_val; |
| result->err = fabs(sum_val); |
| GSL_ERROR ("overflow", GSL_EOVRFLW); |
| } |
| |
| abs_del = fabs(del); |
| max_abs_del = GSL_MAX_DBL(abs_del, max_abs_del); |
| sum_err += 2.0*GSL_DBL_EPSILON*abs_del; |
| |
| an += 1.0; |
| bn += 1.0; |
| n += 1.0; |
| } |
| |
| result->val = sum_val; |
| result->err = sum_err; |
| result->err += abs_del; |
| result->err += 2.0 * GSL_DBL_EPSILON * n * fabs(sum_val); |
| |
| return GSL_SUCCESS; |
| } |
| |
| |
| int |
| gsl_sf_hyperg_1F1_large_b_e(const double a, const double b, const double x, gsl_sf_result * result) |
| { |
| if(fabs(x/b) < 1.0) { |
| const double u = x/b; |
| const double v = 1.0/(1.0-u); |
| const double pre = pow(v,a); |
| const double uv = u*v; |
| const double uv2 = uv*uv; |
| const double t1 = a*(a+1.0)/(2.0*b)*uv2; |
| const double t2a = a*(a+1.0)/(24.0*b*b)*uv2; |
| const double t2b = 12.0 + 16.0*(a+2.0)*uv + 3.0*(a+2.0)*(a+3.0)*uv2; |
| const double t2 = t2a*t2b; |
| result->val = pre * (1.0 - t1 + t2); |
| result->err = pre * GSL_DBL_EPSILON * (1.0 + fabs(t1) + fabs(t2)); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| DOMAIN_ERROR(result); |
| } |
| } |
| |
| |
| int |
| gsl_sf_hyperg_U_large_b_e(const double a, const double b, const double x, |
| gsl_sf_result * result, |
| double * ln_multiplier |
| ) |
| { |
| double N = floor(b); /* b = N + eps */ |
| double eps = b - N; |
| |
| if(fabs(eps) < GSL_SQRT_DBL_EPSILON) { |
| double lnpre_val; |
| double lnpre_err; |
| gsl_sf_result M; |
| if(b > 1.0) { |
| double tmp = (1.0-b)*log(x); |
| gsl_sf_result lg_bm1; |
| gsl_sf_result lg_a; |
| gsl_sf_lngamma_e(b-1.0, &lg_bm1); |
| gsl_sf_lngamma_e(a, &lg_a); |
| lnpre_val = tmp + x + lg_bm1.val - lg_a.val; |
| lnpre_err = lg_bm1.err + lg_a.err + GSL_DBL_EPSILON * (fabs(x) + fabs(tmp)); |
| gsl_sf_hyperg_1F1_large_b_e(1.0-a, 2.0-b, -x, &M); |
| } |
| else { |
| gsl_sf_result lg_1mb; |
| gsl_sf_result lg_1pamb; |
| gsl_sf_lngamma_e(1.0-b, &lg_1mb); |
| gsl_sf_lngamma_e(1.0+a-b, &lg_1pamb); |
| lnpre_val = lg_1mb.val - lg_1pamb.val; |
| lnpre_err = lg_1mb.err + lg_1pamb.err; |
| gsl_sf_hyperg_1F1_large_b_e(a, b, x, &M); |
| } |
| |
| if(lnpre_val > GSL_LOG_DBL_MAX-10.0) { |
| result->val = M.val; |
| result->err = M.err; |
| *ln_multiplier = lnpre_val; |
| GSL_ERROR ("overflow", GSL_EOVRFLW); |
| } |
| else { |
| gsl_sf_result epre; |
| int stat_e = gsl_sf_exp_err_e(lnpre_val, lnpre_err, &epre); |
| result->val = epre.val * M.val; |
| result->err = epre.val * M.err + epre.err * fabs(M.val); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| *ln_multiplier = 0.0; |
| return stat_e; |
| } |
| } |
| else { |
| double omb_lnx = (1.0-b)*log(x); |
| gsl_sf_result lg_1mb; double sgn_1mb; |
| gsl_sf_result lg_1pamb; double sgn_1pamb; |
| gsl_sf_result lg_bm1; double sgn_bm1; |
| gsl_sf_result lg_a; double sgn_a; |
| gsl_sf_result M1, M2; |
| double lnpre1_val, lnpre2_val; |
| double lnpre1_err, lnpre2_err; |
| double sgpre1, sgpre2; |
| gsl_sf_hyperg_1F1_large_b_e( a, b, x, &M1); |
| gsl_sf_hyperg_1F1_large_b_e(1.0-a, 2.0-b, x, &M2); |
| |
| gsl_sf_lngamma_sgn_e(1.0-b, &lg_1mb, &sgn_1mb); |
| gsl_sf_lngamma_sgn_e(1.0+a-b, &lg_1pamb, &sgn_1pamb); |
| |
| gsl_sf_lngamma_sgn_e(b-1.0, &lg_bm1, &sgn_bm1); |
| gsl_sf_lngamma_sgn_e(a, &lg_a, &sgn_a); |
| |
| lnpre1_val = lg_1mb.val - lg_1pamb.val; |
| lnpre1_err = lg_1mb.err + lg_1pamb.err; |
| lnpre2_val = lg_bm1.val - lg_a.val - omb_lnx - x; |
| lnpre2_err = lg_bm1.err + lg_a.err + GSL_DBL_EPSILON * (fabs(omb_lnx)+fabs(x)); |
| sgpre1 = sgn_1mb * sgn_1pamb; |
| sgpre2 = sgn_bm1 * sgn_a; |
| |
| if(lnpre1_val > GSL_LOG_DBL_MAX-10.0 || lnpre2_val > GSL_LOG_DBL_MAX-10.0) { |
| double max_lnpre_val = GSL_MAX(lnpre1_val,lnpre2_val); |
| double max_lnpre_err = GSL_MAX(lnpre1_err,lnpre2_err); |
| double lp1 = lnpre1_val - max_lnpre_val; |
| double lp2 = lnpre2_val - max_lnpre_val; |
| double t1 = sgpre1*exp(lp1); |
| double t2 = sgpre2*exp(lp2); |
| result->val = t1*M1.val + t2*M2.val; |
| result->err = fabs(t1)*M1.err + fabs(t2)*M2.err; |
| result->err += GSL_DBL_EPSILON * exp(max_lnpre_err) * (fabs(t1*M1.val) + fabs(t2*M2.val)); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| *ln_multiplier = max_lnpre_val; |
| GSL_ERROR ("overflow", GSL_EOVRFLW); |
| } |
| else { |
| double t1 = sgpre1*exp(lnpre1_val); |
| double t2 = sgpre2*exp(lnpre2_val); |
| result->val = t1*M1.val + t2*M2.val; |
| result->err = fabs(t1) * M1.err + fabs(t2)*M2.err; |
| result->err += GSL_DBL_EPSILON * (exp(lnpre1_err)*fabs(t1*M1.val) + exp(lnpre2_err)*fabs(t2*M2.val)); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| *ln_multiplier = 0.0; |
| return GSL_SUCCESS; |
| } |
| } |
| } |
| |
| |
| |
| /* [Carlson, p.109] says the error in truncating this asymptotic series |
| * is less than the absolute value of the first neglected term. |
| * |
| * A termination argument is provided, so that the series will |
| * be summed at most up to n=n_trunc. If n_trunc is set negative, |
| * then the series is summed until it appears to start diverging. |
| */ |
| int |
| gsl_sf_hyperg_2F0_series_e(const double a, const double b, const double x, |
| int n_trunc, |
| gsl_sf_result * result |
| ) |
| { |
| const int maxiter = 2000; |
| double an = a; |
| double bn = b; |
| double n = 1.0; |
| double sum = 1.0; |
| double del = 1.0; |
| double abs_del = 1.0; |
| double max_abs_del = 1.0; |
| double last_abs_del = 1.0; |
| |
| while(abs_del/fabs(sum) > GSL_DBL_EPSILON && n < maxiter) { |
| |
| double u = an * (bn/n * x); |
| double abs_u = fabs(u); |
| |
| if(abs_u > 1.0 && (max_abs_del > GSL_DBL_MAX/abs_u)) { |
| result->val = sum; |
| result->err = fabs(sum); |
| GSL_ERROR ("overflow", GSL_EOVRFLW); |
| } |
| |
| del *= u; |
| sum += del; |
| |
| abs_del = fabs(del); |
| |
| if(abs_del > last_abs_del) break; /* series is probably starting to grow */ |
| |
| last_abs_del = abs_del; |
| max_abs_del = GSL_MAX(abs_del, max_abs_del); |
| |
| an += 1.0; |
| bn += 1.0; |
| n += 1.0; |
| |
| if(an == 0.0 || bn == 0.0) break; /* series terminated */ |
| |
| if(n_trunc >= 0 && n >= n_trunc) break; /* reached requested timeout */ |
| } |
| |
| result->val = sum; |
| result->err = GSL_DBL_EPSILON * n + abs_del; |
| if(n >= maxiter) |
| GSL_ERROR ("error", GSL_EMAXITER); |
| else |
| return GSL_SUCCESS; |
| } |
