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- /* specfunc/bessel_Yn.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_gamma.h"
- #include "gsl_sf_psi.h"
- #include "gsl_sf_bessel.h"
- #include "gsl_specfunc__error.h"
- #include "gsl_specfunc__bessel.h"
- #include "gsl_specfunc__bessel_amp_phase.h"
- #include "gsl_specfunc__bessel_olver.h"
- /*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/
- /* assumes n >= 1 */
- static int bessel_Yn_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);
- 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 = -exp(ln_pre1) * sum1 / M_PI;
-
- pre2 = -exp(n*ln_x_2) / M_PI;
- if(fabs(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./k;
- psi_npkp1 += 1./(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 = pre2 * sum2;
- }
- else {
- term2 = 0.0;
- }
- result->val = term1 + term2;
- result->err = 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_Yn_e(int n, const double x, gsl_sf_result * result)
- {
- int sign = 1;
- if(n < 0) {
- /* reduce to case n >= 0 */
- n = -n;
- if(GSL_IS_ODD(n)) sign = -1;
- }
- /* CHECK_POINTER(result) */
- if(n == 0) {
- int status = gsl_sf_bessel_Y0_e(x, result);
- result->val *= sign;
- return status;
- }
- else if(n == 1) {
- int status = gsl_sf_bessel_Y1_e(x, result);
- result->val *= sign;
- return status;
- }
- else {
- if(x <= 0.0) {
- DOMAIN_ERROR(result);
- }
- if(x < 5.0) {
- int status = bessel_Yn_small_x(n, x, result);
- result->val *= sign;
- return status;
- }
- else if(GSL_ROOT3_DBL_EPSILON * x > (n*n + 1.0)) {
- int status = gsl_sf_bessel_Ynu_asympx_e((double)n, x, result);
- result->val *= sign;
- return status;
- }
- else if(n > 50) {
- int status = gsl_sf_bessel_Ynu_asymp_Olver_e((double)n, x, result);
- result->val *= sign;
- return status;
- }
- else {
- double two_over_x = 2.0/x;
- gsl_sf_result r_by;
- gsl_sf_result r_bym;
- int stat_1 = gsl_sf_bessel_Y1_e(x, &r_by);
- int stat_0 = gsl_sf_bessel_Y0_e(x, &r_bym);
- double bym = r_bym.val;
- double by = r_by.val;
- double byp;
- int j;
- for(j=1; j<n; j++) {
- byp = j*two_over_x*by - bym;
- bym = by;
- by = byp;
- }
- result->val = sign * by;
- result->err = fabs(result->val) * (fabs(r_by.err/r_by.val) + fabs(r_bym.err/r_bym.val));
- result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return GSL_ERROR_SELECT_2(stat_1, stat_0);
- }
- }
- }
- int
- gsl_sf_bessel_Yn_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 ("error", GSL_EDOM);
- }
- else {
- gsl_sf_result r_Ynm1;
- gsl_sf_result r_Yn;
- int stat_nm1 = gsl_sf_bessel_Yn_e(nmin, x, &r_Ynm1);
- int stat_n = gsl_sf_bessel_Yn_e(nmin+1, x, &r_Yn);
- double Ynp1;
- double Yn = r_Yn.val;
- double Ynm1 = r_Ynm1.val;
- int n;
- int stat = GSL_ERROR_SELECT_2(stat_nm1, stat_n);
- if(stat == GSL_SUCCESS) {
- for(n=nmin+1; n<=nmax+1; n++) {
- result_array[n-nmin-1] = Ynm1;
- Ynp1 = -Ynm1 + 2.0*n/x * Yn;
- Ynm1 = Yn;
- Yn = Ynp1;
- }
- }
- else {
- for(n=nmin; n<=nmax; n++) {
- result_array[n-nmin] = 0.0;
- }
- }
- return stat;
- }
- }
- /*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
- #include "gsl_specfunc__eval.h"
- double gsl_sf_bessel_Yn(const int n, const double x)
- {
- EVAL_RESULT(gsl_sf_bessel_Yn_e(n, x, &result));
- }
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