| /* specfunc/bessel_Knu.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 "error.h" |
| |
| #include "bessel.h" |
| #include "bessel_temme.h" |
| |
| /*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/ |
| |
| int |
| gsl_sf_bessel_Knu_scaled_e(const double nu, const double x, gsl_sf_result * result) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| if(x <= 0.0 || nu < 0.0) { |
| DOMAIN_ERROR(result); |
| } |
| else { |
| int N = (int)(nu + 0.5); |
| double mu = nu - N; /* -1/2 <= mu <= 1/2 */ |
| double K_mu, K_mup1, Kp_mu; |
| double K_nu, K_nup1, K_num1; |
| int n; |
| |
| if(x < 2.0) { |
| gsl_sf_bessel_K_scaled_temme(mu, x, &K_mu, &K_mup1, &Kp_mu); |
| } |
| else { |
| gsl_sf_bessel_K_scaled_steed_temme_CF2(mu, x, &K_mu, &K_mup1, &Kp_mu); |
| } |
| |
| /* recurse forward to obtain K_num1, K_nu */ |
| K_nu = K_mu; |
| K_nup1 = K_mup1; |
| |
| for(n=0; n<N; n++) { |
| K_num1 = K_nu; |
| K_nu = K_nup1; |
| K_nup1 = 2.0*(mu+n+1)/x * K_nu + K_num1; |
| } |
| |
| result->val = K_nu; |
| result->err = 2.0 * GSL_DBL_EPSILON * (N + 4.0) * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| } |
| |
| |
| int |
| gsl_sf_bessel_Knu_e(const double nu, const double x, gsl_sf_result * result) |
| { |
| gsl_sf_result b; |
| int stat_K = gsl_sf_bessel_Knu_scaled_e(nu, x, &b); |
| int stat_e = gsl_sf_exp_mult_err_e(-x, 0.0, b.val, b.err, result); |
| return GSL_ERROR_SELECT_2(stat_e, stat_K); |
| } |
| |
| |
| int |
| gsl_sf_bessel_lnKnu_e(const double nu, const double x, gsl_sf_result * result) |
| { |
| /* CHECK_POINTER(result) */ |
| |
| if(x <= 0.0 || nu < 0.0) { |
| DOMAIN_ERROR(result); |
| } |
| else if(nu == 0.0) { |
| gsl_sf_result K_scaled; |
| /* This cannot underflow, and |
| * it will not throw GSL_EDOM |
| * since that is already checked. |
| */ |
| gsl_sf_bessel_K0_scaled_e(x, &K_scaled); |
| result->val = -x + log(fabs(K_scaled.val)); |
| result->err = GSL_DBL_EPSILON * fabs(x) + fabs(K_scaled.err/K_scaled.val); |
| result->err += GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else if(x < 2.0 && nu > 1.0) { |
| /* Make use of the inequality |
| * Knu(x) <= 1/2 (2/x)^nu Gamma(nu), |
| * which follows from the integral representation |
| * [Abramowitz+Stegun, 9.6.23 (2)]. With this |
| * we decide whether or not there is an overflow |
| * problem because x is small. |
| */ |
| double ln_bound; |
| gsl_sf_result lg_nu; |
| gsl_sf_lngamma_e(nu, &lg_nu); |
| ln_bound = -M_LN2 - nu*log(0.5*x) + lg_nu.val; |
| if(ln_bound > GSL_LOG_DBL_MAX - 20.0) { |
| /* x must be very small or nu very large (or both). |
| */ |
| double xi = 0.25*x*x; |
| double sum = 1.0 - xi/(nu-1.0); |
| if(nu > 2.0) sum += (xi/(nu-1.0)) * (xi/(nu-2.0)); |
| result->val = ln_bound + log(sum); |
| result->err = lg_nu.err; |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| /* can drop-through here */ |
| } |
| |
| |
| { |
| /* We passed the above tests, so no problem. |
| * Evaluate as usual. Note the possible drop-through |
| * in the above code! |
| */ |
| gsl_sf_result K_scaled; |
| gsl_sf_bessel_Knu_scaled_e(nu, x, &K_scaled); |
| result->val = -x + log(fabs(K_scaled.val)); |
| result->err = GSL_DBL_EPSILON * fabs(x) + fabs(K_scaled.err/K_scaled.val); |
| result->err += GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| } |
| |
| |
| /*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/ |
| |
| #include "eval.h" |
| |
| double gsl_sf_bessel_Knu_scaled(const double nu, const double x) |
| { |
| EVAL_RESULT(gsl_sf_bessel_Knu_scaled_e(nu, x, &result)); |
| } |
| |
| double gsl_sf_bessel_Knu(const double nu, const double x) |
| { |
| EVAL_RESULT(gsl_sf_bessel_Knu_e(nu, x, &result)); |
| } |
| |
| double gsl_sf_bessel_lnKnu(const double nu, const double x) |
| { |
| EVAL_RESULT(gsl_sf_bessel_lnKnu_e(nu, x, &result)); |
| } |