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- /* specfunc/bessel_k.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 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 "gsl__config.h"
- #include "gsl_math.h"
- #include "gsl_errno.h"
- #include "gsl_sf_pow_int.h"
- #include "gsl_sf_gamma.h"
- #include "gsl_sf_bessel.h"
- #include "gsl_specfunc__error.h"
- #include "gsl_specfunc__check.h"
- #include "gsl_specfunc__bessel.h"
- /*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/
- /* [Abramowitz+Stegun, 10.2.4 + 10.2.6]
- * with lmax=15, precision ~ 15D for x < 3
- *
- * assumes l >= 1
- */
- static int bessel_kl_scaled_small_x(int l, const double x, gsl_sf_result * result)
- {
- gsl_sf_result num_fact;
- double den = gsl_sf_pow_int(x, l+1);
- int stat_df = gsl_sf_doublefact_e((unsigned int) (2*l-1), &num_fact);
- if(stat_df != GSL_SUCCESS || den == 0.0) {
- OVERFLOW_ERROR(result);
- }
- else {
- const int lmax = 50;
- gsl_sf_result ipos_term;
- double ineg_term;
- double sgn = (GSL_IS_ODD(l) ? -1.0 : 1.0);
- double ex = exp(x);
- double t = 0.5*x*x;
- double sum = 1.0;
- double t_coeff = 1.0;
- double t_power = 1.0;
- double delta;
- int stat_il;
- int i;
- for(i=1; i<lmax; i++) {
- t_coeff /= i*(2*(i-l) - 1);
- t_power *= t;
- delta = t_power*t_coeff;
- sum += delta;
- if(fabs(delta/sum) < GSL_DBL_EPSILON) break;
- }
- stat_il = gsl_sf_bessel_il_scaled_e(l, x, &ipos_term);
- ineg_term = sgn * num_fact.val/den * sum;
- result->val = -sgn * 0.5*M_PI * (ex*ipos_term.val - ineg_term);
- result->val *= ex;
- result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return stat_il;
- }
- }
- /*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
- int gsl_sf_bessel_k0_scaled_e(const double x, gsl_sf_result * result)
- {
- /* CHECK_POINTER(result) */
- if(x <= 0.0) {
- DOMAIN_ERROR(result);
- }
- else {
- result->val = M_PI/(2.0*x);
- result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- CHECK_UNDERFLOW(result);
- return GSL_SUCCESS;
- }
- }
- int gsl_sf_bessel_k1_scaled_e(const double x, gsl_sf_result * result)
- {
- /* CHECK_POINTER(result) */
- if(x <= 0.0) {
- DOMAIN_ERROR(result);
- }
- else if(x < (M_SQRTPI+1.0)/(M_SQRT2*GSL_SQRT_DBL_MAX)) {
- OVERFLOW_ERROR(result);
- }
- else {
- result->val = M_PI/(2.0*x) * (1.0 + 1.0/x);
- result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- CHECK_UNDERFLOW(result);
- return GSL_SUCCESS;
- }
- }
- int gsl_sf_bessel_k2_scaled_e(const double x, gsl_sf_result * result)
- {
- /* CHECK_POINTER(result) */
- if(x <= 0.0) {
- DOMAIN_ERROR(result);
- }
- else if(x < 2.0/GSL_ROOT3_DBL_MAX) {
- OVERFLOW_ERROR(result);
- }
- else {
- result->val = M_PI/(2.0*x) * (1.0 + 3.0/x * (1.0 + 1.0/x));
- result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- CHECK_UNDERFLOW(result);
- return GSL_SUCCESS;
- }
- }
- int gsl_sf_bessel_kl_scaled_e(int l, const double x, gsl_sf_result * result)
- {
- if(l < 0 || x <= 0.0) {
- DOMAIN_ERROR(result);
- }
- else if(l == 0) {
- return gsl_sf_bessel_k0_scaled_e(x, result);
- }
- else if(l == 1) {
- return gsl_sf_bessel_k1_scaled_e(x, result);
- }
- else if(l == 2) {
- return gsl_sf_bessel_k2_scaled_e(x, result);
- }
- else if(x < 3.0) {
- return bessel_kl_scaled_small_x(l, x, result);
- }
- else if(GSL_ROOT3_DBL_EPSILON * x > (l*l + l + 1)) {
- int status = gsl_sf_bessel_Knu_scaled_asympx_e(l + 0.5, x, result);
- double pre = sqrt((0.5*M_PI)/x);
- result->val *= pre;
- result->err *= pre;
- return status;
- }
- else if(GSL_MIN(0.29/(l*l+1.0), 0.5/(l*l+1.0+x*x)) < GSL_ROOT3_DBL_EPSILON) {
- int status = gsl_sf_bessel_Knu_scaled_asymp_unif_e(l + 0.5, x, result);
- double pre = sqrt((0.5*M_PI)/x);
- result->val *= pre;
- result->err *= pre;
- return status;
- }
- else {
- /* recurse upward */
- gsl_sf_result r_bk;
- gsl_sf_result r_bkm;
- int stat_1 = gsl_sf_bessel_k1_scaled_e(x, &r_bk);
- int stat_0 = gsl_sf_bessel_k0_scaled_e(x, &r_bkm);
- double bkp;
- double bk = r_bk.val;
- double bkm = r_bkm.val;
- int j;
- for(j=1; j<l; j++) {
- bkp = (2*j+1)/x*bk + bkm;
- bkm = bk;
- bk = bkp;
- }
- result->val = bk;
- result->err = fabs(bk) * (fabs(r_bk.err/r_bk.val) + fabs(r_bkm.err/r_bkm.val));
- result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return GSL_ERROR_SELECT_2(stat_1, stat_0);
- }
- }
- int
- gsl_sf_bessel_kl_scaled_array(const int lmax, const double x, double * result_array)
- {
- if(lmax < 0 || x <= 0.0) {
- GSL_ERROR("domain error", GSL_EDOM);
- } else if (lmax == 0) {
- gsl_sf_result result;
- int stat = gsl_sf_bessel_k0_scaled_e(x, &result);
- result_array[0] = result.val;
- return stat;
- } else {
- int ell;
- double kellp1, kell, kellm1;
- gsl_sf_result r_kell;
- gsl_sf_result r_kellm1;
- gsl_sf_bessel_k1_scaled_e(x, &r_kell);
- gsl_sf_bessel_k0_scaled_e(x, &r_kellm1);
- kell = r_kell.val;
- kellm1 = r_kellm1.val;
- result_array[0] = kellm1;
- result_array[1] = kell;
- for(ell = 1; ell < lmax; ell++) {
- kellp1 = (2*ell+1)/x * kell + kellm1;
- result_array[ell+1] = kellp1;
- kellm1 = kell;
- kell = kellp1;
- }
- return GSL_SUCCESS;
- }
- }
- /*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
- #include "gsl_specfunc__eval.h"
- double gsl_sf_bessel_k0_scaled(const double x)
- {
- EVAL_RESULT(gsl_sf_bessel_k0_scaled_e(x, &result));
- }
- double gsl_sf_bessel_k1_scaled(const double x)
- {
- EVAL_RESULT(gsl_sf_bessel_k1_scaled_e(x, &result));
- }
- double gsl_sf_bessel_k2_scaled(const double x)
- {
- EVAL_RESULT(gsl_sf_bessel_k2_scaled_e(x, &result));
- }
- double gsl_sf_bessel_kl_scaled(const int l, const double x)
- {
- EVAL_RESULT(gsl_sf_bessel_kl_scaled_e(l, x, &result));
- }
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