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/* specfunc/bessel_Kn.c
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*
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* Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman
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*
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* This program is free software; you can redistribute it and/or modify
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* it under the terms of the GNU General Public License as published by
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* the Free Software Foundation; either version 3 of the License, or (at
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* your option) any later version.
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*
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* This program is distributed in the hope that it will be useful, but
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* WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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* General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License
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* along with this program; if not, write to the Free Software
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* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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*/
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/* Author: G. Jungman */
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#include <config.h>
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#include <gsl/gsl_math.h>
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#include <gsl/gsl_errno.h>
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#include <gsl/gsl_sf_gamma.h>
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#include <gsl/gsl_sf_psi.h>
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#include <gsl/gsl_sf_bessel.h>
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#include "error.h"
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#include "bessel.h"
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/*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/
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/* [Abramowitz+Stegun, 9.6.11]
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* assumes n >= 1
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*/
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static
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int
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bessel_Kn_scaled_small_x(const int n, const double x, gsl_sf_result * result)
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{
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int k;
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double y = 0.25 * x * x;
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double ln_x_2 = log(0.5*x);
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double ex = exp(x);
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gsl_sf_result ln_nm1_fact;
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double k_term;
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double term1, sum1, ln_pre1;
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double term2, sum2, pre2;
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gsl_sf_lnfact_e((unsigned int)(n-1), &ln_nm1_fact);
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ln_pre1 = -n*ln_x_2 + ln_nm1_fact.val;
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if(ln_pre1 > GSL_LOG_DBL_MAX - 3.0) GSL_ERROR ("error", GSL_EOVRFLW);
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sum1 = 1.0;
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k_term = 1.0;
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for(k=1; k<=n-1; k++) {
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k_term *= -y/(k * (n-k));
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sum1 += k_term;
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}
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term1 = 0.5 * exp(ln_pre1) * sum1;
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pre2 = 0.5 * exp(n*ln_x_2);
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if(pre2 > 0.0) {
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const int KMAX = 20;
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gsl_sf_result psi_n;
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gsl_sf_result npk_fact;
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double yk = 1.0;
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double k_fact = 1.0;
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double psi_kp1 = -M_EULER;
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double psi_npkp1;
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gsl_sf_psi_int_e(n, &psi_n);
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gsl_sf_fact_e((unsigned int)n, &npk_fact);
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psi_npkp1 = psi_n.val + 1.0/n;
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sum2 = (psi_kp1 + psi_npkp1 - 2.0*ln_x_2)/npk_fact.val;
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for(k=1; k
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psi_kp1 += 1.0/k;
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psi_npkp1 += 1.0/(n+k);
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k_fact *= k;
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npk_fact.val *= n+k;
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yk *= y;
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k_term = yk*(psi_kp1 + psi_npkp1 - 2.0*ln_x_2)/(k_fact*npk_fact.val);
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sum2 += k_term;
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}
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term2 = ( GSL_IS_ODD(n) ? -1.0 : 1.0 ) * pre2 * sum2;
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}
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else {
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term2 = 0.0;
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}
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result->val = ex * (term1 + term2);
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result->err = ex * GSL_DBL_EPSILON * (fabs(ln_pre1)*fabs(term1) + fabs(term2));
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result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
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return GSL_SUCCESS;
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}
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/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
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int gsl_sf_bessel_Kn_scaled_e(int n, const double x, gsl_sf_result * result)
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{
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n = abs(n); /* K(-n, z) = K(n, z) */
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/* CHECK_POINTER(result) */
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if(x <= 0.0) {
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DOMAIN_ERROR(result);
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}
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else if(n == 0) {
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return gsl_sf_bessel_K0_scaled_e(x, result);
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}
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else if(n == 1) {
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return gsl_sf_bessel_K1_scaled_e(x, result);
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}
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else if(x <= 5.0) {
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return bessel_Kn_scaled_small_x(n, x, result);
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}
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else if(GSL_ROOT3_DBL_EPSILON * x > 0.25 * (n*n + 1)) {
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return gsl_sf_bessel_Knu_scaled_asympx_e((double)n, x, result);
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}
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else if(GSL_MIN(0.29/(n*n), 0.5/(n*n + x*x)) < GSL_ROOT3_DBL_EPSILON) {
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return gsl_sf_bessel_Knu_scaled_asymp_unif_e((double)n, x, result);
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}
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else {
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/* Upward recurrence. [Gradshteyn + Ryzhik, 8.471.1] */
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double two_over_x = 2.0/x;
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gsl_sf_result r_b_jm1;
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gsl_sf_result r_b_j;
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int stat_0 = gsl_sf_bessel_K0_scaled_e(x, &r_b_jm1);
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int stat_1 = gsl_sf_bessel_K1_scaled_e(x, &r_b_j);
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double b_jm1 = r_b_jm1.val;
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double b_j = r_b_j.val;
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double b_jp1;
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int j;
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for(j=1; j
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b_jp1 = b_jm1 + j * two_over_x * b_j;
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b_jm1 = b_j;
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b_j = b_jp1;
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}
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result->val = b_j;
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result->err = n * (fabs(b_j) * (fabs(r_b_jm1.err/r_b_jm1.val) + fabs(r_b_j.err/r_b_j.val)));
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result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
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return GSL_ERROR_SELECT_2(stat_0, stat_1);
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}
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}
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int gsl_sf_bessel_Kn_e(const int n, const double x, gsl_sf_result * result)
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{
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const int status = gsl_sf_bessel_Kn_scaled_e(n, x, result);
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const double ex = exp(-x);
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result->val *= ex;
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result->err *= ex;
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result->err += x * GSL_DBL_EPSILON * fabs(result->val);
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return status;
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}
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int gsl_sf_bessel_Kn_scaled_array(const int nmin, const int nmax, const double x, double * result_array)
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{
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/* CHECK_POINTER(result_array) */
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if(nmin < 0 || nmax < nmin || x <= 0.0) {
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int j;
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for(j=0; j<=nmax-nmin; j++) result_array[j] = 0.0;
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GSL_ERROR ("domain error", GSL_EDOM);
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}
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else if(nmax == 0) {
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gsl_sf_result b;
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int stat = gsl_sf_bessel_K0_scaled_e(x, &b);
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result_array[0] = b.val;
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return stat;
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}
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else {
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double two_over_x = 2.0/x;
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gsl_sf_result r_Knm1;
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gsl_sf_result r_Kn;
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int stat_0 = gsl_sf_bessel_Kn_scaled_e(nmin, x, &r_Knm1);
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int stat_1 = gsl_sf_bessel_Kn_scaled_e(nmin+1, x, &r_Kn);
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int stat = GSL_ERROR_SELECT_2(stat_0, stat_1);
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double Knp1;
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double Kn = r_Kn.val;
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double Knm1 = r_Knm1.val;
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int n;
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for(n=nmin+1; n<=nmax+1; n++) {
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if(Knm1 < GSL_DBL_MAX) {
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result_array[n-1-nmin] = Knm1;
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Knp1 = Knm1 + n * two_over_x * Kn;
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Knm1 = Kn;
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Kn = Knp1;
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}
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else {
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/* Overflow. Set the rest of the elements to
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* zero and bug out.
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* FIXME: Note: this relies on the convention
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* that the test x < DBL_MIN fails for x not
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* a number. This may be only an IEEE convention,
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* so the portability is unclear.
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*/
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int j;
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for(j=n; j<=nmax+1; j++) result_array[j-1-nmin] = 0.0;
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GSL_ERROR ("overflow", GSL_EOVRFLW);
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}
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}
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return stat;
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}
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}
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int
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gsl_sf_bessel_Kn_array(const int nmin, const int nmax, const double x, double * result_array)
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{
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int status = gsl_sf_bessel_Kn_scaled_array(nmin, nmax, x, result_array);
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double ex = exp(-x);
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int i;
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for(i=0; i<=nmax-nmin; i++) result_array[i] *= ex;
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return status;
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}
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/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
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#include "eval.h"
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double gsl_sf_bessel_Kn_scaled(const int n, const double x)
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{
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EVAL_RESULT(gsl_sf_bessel_Kn_scaled_e(n, x, &result));
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}
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double gsl_sf_bessel_Kn(const int n, const double x)
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{
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EVAL_RESULT(gsl_sf_bessel_Kn_e(n, x, &result));
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}
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