| /* specfunc/bessel_Kn.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_gamma.h> |
| #include <gsl/gsl_sf_psi.h> |
| #include <gsl/gsl_sf_bessel.h> |
| |
| #include "error.h" |
| |
| #include "bessel.h" |
| |
| /*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/ |
| |
| /* [Abramowitz+Stegun, 9.6.11] |
| * assumes n >= 1 |
| */ |
| static |
| int |
| bessel_Kn_scaled_small_x(const int n, const double x, gsl_sf_result * result) |
| { |
| int k; |
| double y = 0.25 * x * x; |
| double ln_x_2 = log(0.5*x); |
| double ex = exp(x); |
| gsl_sf_result ln_nm1_fact; |
| double k_term; |
| double term1, sum1, ln_pre1; |
| double term2, sum2, pre2; |
| |
| gsl_sf_lnfact_e((unsigned int)(n-1), &ln_nm1_fact); |
| |
| ln_pre1 = -n*ln_x_2 + ln_nm1_fact.val; |
| if(ln_pre1 > GSL_LOG_DBL_MAX - 3.0) GSL_ERROR ("error", GSL_EOVRFLW); |
| |
| sum1 = 1.0; |
| k_term = 1.0; |
| for(k=1; k<=n-1; k++) { |
| k_term *= -y/(k * (n-k)); |
| sum1 += k_term; |
| } |
| term1 = 0.5 * exp(ln_pre1) * sum1; |
| |
| pre2 = 0.5 * exp(n*ln_x_2); |
| if(pre2 > 0.0) { |
| const int KMAX = 20; |
| gsl_sf_result psi_n; |
| gsl_sf_result npk_fact; |
| double yk = 1.0; |
| double k_fact = 1.0; |
| double psi_kp1 = -M_EULER; |
| double psi_npkp1; |
| gsl_sf_psi_int_e(n, &psi_n); |
| gsl_sf_fact_e((unsigned int)n, &npk_fact); |
| psi_npkp1 = psi_n.val + 1.0/n; |
| sum2 = (psi_kp1 + psi_npkp1 - 2.0*ln_x_2)/npk_fact.val; |
| for(k=1; k<KMAX; k++) { |
| psi_kp1 += 1.0/k; |
| psi_npkp1 += 1.0/(n+k); |
| k_fact *= k; |
| npk_fact.val *= n+k; |
| yk *= y; |
| k_term = yk*(psi_kp1 + psi_npkp1 - 2.0*ln_x_2)/(k_fact*npk_fact.val); |
| sum2 += k_term; |
| } |
| term2 = ( GSL_IS_ODD(n) ? -1.0 : 1.0 ) * pre2 * sum2; |
| } |
| else { |
| term2 = 0.0; |
| } |
| |
| result->val = ex * (term1 + term2); |
| result->err = ex * GSL_DBL_EPSILON * (fabs(ln_pre1)*fabs(term1) + fabs(term2)); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| |
| return GSL_SUCCESS; |
| } |
| |
| |
| /*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/ |
| |
| int gsl_sf_bessel_Kn_scaled_e(int n, const double x, gsl_sf_result * result) |
| { |
| n = abs(n); /* K(-n, z) = K(n, z) */ |
| |
| /* CHECK_POINTER(result) */ |
| |
| if(x <= 0.0) { |
| DOMAIN_ERROR(result); |
| } |
| else if(n == 0) { |
| return gsl_sf_bessel_K0_scaled_e(x, result); |
| } |
| else if(n == 1) { |
| return gsl_sf_bessel_K1_scaled_e(x, result); |
| } |
| else if(x <= 5.0) { |
| return bessel_Kn_scaled_small_x(n, x, result); |
| } |
| else if(GSL_ROOT3_DBL_EPSILON * x > 0.25 * (n*n + 1)) { |
| return gsl_sf_bessel_Knu_scaled_asympx_e((double)n, x, result); |
| } |
| else if(GSL_MIN(0.29/(n*n), 0.5/(n*n + x*x)) < GSL_ROOT3_DBL_EPSILON) { |
| return gsl_sf_bessel_Knu_scaled_asymp_unif_e((double)n, x, result); |
| } |
| else { |
| /* Upward recurrence. [Gradshteyn + Ryzhik, 8.471.1] */ |
| double two_over_x = 2.0/x; |
| gsl_sf_result r_b_jm1; |
| gsl_sf_result r_b_j; |
| int stat_0 = gsl_sf_bessel_K0_scaled_e(x, &r_b_jm1); |
| int stat_1 = gsl_sf_bessel_K1_scaled_e(x, &r_b_j); |
| double b_jm1 = r_b_jm1.val; |
| double b_j = r_b_j.val; |
| double b_jp1; |
| int j; |
| |
| for(j=1; j<n; j++) { |
| b_jp1 = b_jm1 + j * two_over_x * b_j; |
| b_jm1 = b_j; |
| b_j = b_jp1; |
| } |
| |
| result->val = b_j; |
| result->err = n * (fabs(b_j) * (fabs(r_b_jm1.err/r_b_jm1.val) + fabs(r_b_j.err/r_b_j.val))); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| |
| return GSL_ERROR_SELECT_2(stat_0, stat_1); |
| } |
| } |
| |
| |
| int gsl_sf_bessel_Kn_e(const int n, const double x, gsl_sf_result * result) |
| { |
| const int status = gsl_sf_bessel_Kn_scaled_e(n, x, result); |
| const double ex = exp(-x); |
| result->val *= ex; |
| result->err *= ex; |
| result->err += x * GSL_DBL_EPSILON * fabs(result->val); |
| return status; |
| } |
| |
| |
| int gsl_sf_bessel_Kn_scaled_array(const int nmin, const int nmax, const double x, double * result_array) |
| { |
| /* CHECK_POINTER(result_array) */ |
| |
| if(nmin < 0 || nmax < nmin || x <= 0.0) { |
| int j; |
| for(j=0; j<=nmax-nmin; j++) result_array[j] = 0.0; |
| GSL_ERROR ("domain error", GSL_EDOM); |
| } |
| else if(nmax == 0) { |
| gsl_sf_result b; |
| int stat = gsl_sf_bessel_K0_scaled_e(x, &b); |
| result_array[0] = b.val; |
| return stat; |
| } |
| else { |
| double two_over_x = 2.0/x; |
| gsl_sf_result r_Knm1; |
| gsl_sf_result r_Kn; |
| int stat_0 = gsl_sf_bessel_Kn_scaled_e(nmin, x, &r_Knm1); |
| int stat_1 = gsl_sf_bessel_Kn_scaled_e(nmin+1, x, &r_Kn); |
| int stat = GSL_ERROR_SELECT_2(stat_0, stat_1); |
| double Knp1; |
| double Kn = r_Kn.val; |
| double Knm1 = r_Knm1.val; |
| int n; |
| |
| for(n=nmin+1; n<=nmax+1; n++) { |
| if(Knm1 < GSL_DBL_MAX) { |
| result_array[n-1-nmin] = Knm1; |
| Knp1 = Knm1 + n * two_over_x * Kn; |
| Knm1 = Kn; |
| Kn = Knp1; |
| } |
| else { |
| /* Overflow. Set the rest of the elements to |
| * zero and bug out. |
| * FIXME: Note: this relies on the convention |
| * that the test x < DBL_MIN fails for x not |
| * a number. This may be only an IEEE convention, |
| * so the portability is unclear. |
| */ |
| int j; |
| for(j=n; j<=nmax+1; j++) result_array[j-1-nmin] = 0.0; |
| GSL_ERROR ("overflow", GSL_EOVRFLW); |
| } |
| } |
| |
| return stat; |
| } |
| } |
| |
| |
| int |
| gsl_sf_bessel_Kn_array(const int nmin, const int nmax, const double x, double * result_array) |
| { |
| int status = gsl_sf_bessel_Kn_scaled_array(nmin, nmax, x, result_array); |
| double ex = exp(-x); |
| int i; |
| for(i=0; i<=nmax-nmin; i++) result_array[i] *= ex; |
| return status; |
| } |
| |
| |
| /*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/ |
| |
| #include "eval.h" |
| |
| double gsl_sf_bessel_Kn_scaled(const int n, const double x) |
| { |
| EVAL_RESULT(gsl_sf_bessel_Kn_scaled_e(n, x, &result)); |
| } |
| |
| double gsl_sf_bessel_Kn(const int n, const double x) |
| { |
| EVAL_RESULT(gsl_sf_bessel_Kn_e(n, x, &result)); |
| } |