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- /* specfunc/bessel_y.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_trig.h"
- #include "gsl_sf_bessel.h"
- #include "gsl_specfunc__error.h"
- #include "gsl_specfunc__bessel.h"
- #include "gsl_specfunc__bessel_olver.h"
- /*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/
- /* [Abramowitz+Stegun, 10.1.3]
- * with lmax=15, precision ~ 15D for x < 3
- *
- * checked OK [GJ] Wed May 13 15:41:25 MDT 1998
- */
- static int bessel_yl_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(2*l-1, &num_fact);
- if(stat_df != GSL_SUCCESS || den == 0.0) {
- OVERFLOW_ERROR(result);
- }
- else {
- const int lmax = 200;
- double t = -0.5*x*x;
- double sum = 1.0;
- double t_coeff = 1.0;
- double t_power = 1.0;
- double delta;
- 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) < 0.5*GSL_DBL_EPSILON) break;
- }
- result->val = -num_fact.val/den * sum;
- result->err = GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- }
- /*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
- int gsl_sf_bessel_y0_e(const double x, gsl_sf_result * result)
- {
- /* CHECK_POINTER(result) */
- if(x <= 0.0) {
- DOMAIN_ERROR(result);
- }
- else if(1.0/GSL_DBL_MAX > 0.0 && x < 1.0/GSL_DBL_MAX) {
- OVERFLOW_ERROR(result);
- }
- else {
- gsl_sf_result cos_result;
- const int stat = gsl_sf_cos_e(x, &cos_result);
- result->val = -cos_result.val/x;
- result->err = fabs(cos_result.err/x);
- result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
- return stat;
- }
- }
- int gsl_sf_bessel_y1_e(const double x, gsl_sf_result * result)
- {
- /* CHECK_POINTER(result) */
- if(x <= 0.0) {
- DOMAIN_ERROR(result);
- }
- else if(x < 1.0/GSL_SQRT_DBL_MAX) {
- OVERFLOW_ERROR(result);
- }
- else if(x < 0.25) {
- const double y = x*x;
- const double c1 = 1.0/2.0;
- const double c2 = -1.0/8.0;
- const double c3 = 1.0/144.0;
- const double c4 = -1.0/5760.0;
- const double c5 = 1.0/403200.0;
- const double c6 = -1.0/43545600.0;
- const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*(c5 + y*c6)))));
- result->val = -sum/y;
- result->err = GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- else {
- gsl_sf_result cos_result;
- gsl_sf_result sin_result;
- const int stat_cos = gsl_sf_cos_e(x, &cos_result);
- const int stat_sin = gsl_sf_sin_e(x, &sin_result);
- const double cx = cos_result.val;
- const double sx = sin_result.val;
- result->val = -(cx/x + sx)/x;
- result->err = (fabs(cos_result.err/x) + sin_result.err)/fabs(x);
- result->err += GSL_DBL_EPSILON * (fabs(sx/x) + fabs(cx/(x*x)));
- return GSL_ERROR_SELECT_2(stat_cos, stat_sin);
- }
- }
- int gsl_sf_bessel_y2_e(const double x, gsl_sf_result * result)
- {
- /* CHECK_POINTER(result) */
- if(x <= 0.0) {
- DOMAIN_ERROR(result);
- }
- else if(x < 1.0/GSL_ROOT3_DBL_MAX) {
- OVERFLOW_ERROR(result);
- }
- else if(x < 0.5) {
- const double y = x*x;
- const double c1 = 1.0/6.0;
- const double c2 = 1.0/24.0;
- const double c3 = -1.0/144.0;
- const double c4 = 1.0/3456.0;
- const double c5 = -1.0/172800.0;
- const double c6 = 1.0/14515200.0;
- const double c7 = -1.0/1828915200.0;
- const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*(c5 + y*(c6 + y*c7))))));
- result->val = -3.0/(x*x*x) * sum;
- result->err = GSL_DBL_EPSILON * fabs(result->val);
- return GSL_SUCCESS;
- }
- else {
- gsl_sf_result cos_result;
- gsl_sf_result sin_result;
- const int stat_cos = gsl_sf_cos_e(x, &cos_result);
- const int stat_sin = gsl_sf_sin_e(x, &sin_result);
- const double sx = sin_result.val;
- const double cx = cos_result.val;
- const double a = 3.0/(x*x);
- result->val = (1.0 - a)/x * cx - a * sx;
- result->err = cos_result.err * fabs((1.0 - a)/x) + sin_result.err * fabs(a);
- result->err += GSL_DBL_EPSILON * (fabs(cx/x) + fabs(sx/(x*x)));
- return GSL_ERROR_SELECT_2(stat_cos, stat_sin);
- }
- }
- int gsl_sf_bessel_yl_e(int l, const double x, gsl_sf_result * result)
- {
- /* CHECK_POINTER(result) */
- if(l < 0 || x <= 0.0) {
- DOMAIN_ERROR(result);
- }
- else if(l == 0) {
- return gsl_sf_bessel_y0_e(x, result);
- }
- else if(l == 1) {
- return gsl_sf_bessel_y1_e(x, result);
- }
- else if(l == 2) {
- return gsl_sf_bessel_y2_e(x, result);
- }
- else if(x < 3.0) {
- return bessel_yl_small_x(l, x, result);
- }
- else if(GSL_ROOT3_DBL_EPSILON * x > (l*l + l + 1.0)) {
- int status = gsl_sf_bessel_Ynu_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(l > 40) {
- int status = gsl_sf_bessel_Ynu_asymp_Olver_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_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<l; j++) {
- byp = (2*j+1)/x*by - bym;
- bym = by;
- by = byp;
- }
- result->val = by;
- result->err = fabs(result->val) * (GSL_DBL_EPSILON + fabs(r_by.err/r_by.val) + fabs(r_bym.err/r_bym.val));
- return GSL_ERROR_SELECT_2(stat_1, stat_0);
- }
- }
- int gsl_sf_bessel_yl_array(const int lmax, const double x, double * result_array)
- {
- /* CHECK_POINTER(result_array) */
- if(lmax < 0 || x <= 0.0) {
- GSL_ERROR ("error", GSL_EDOM);
- } else if (lmax == 0) {
- gsl_sf_result result;
- int stat = gsl_sf_bessel_y0_e(x, &result);
- result_array[0] = result.val;
- return stat;
- } else {
- gsl_sf_result r_yell;
- gsl_sf_result r_yellm1;
- int stat_1 = gsl_sf_bessel_y1_e(x, &r_yell);
- int stat_0 = gsl_sf_bessel_y0_e(x, &r_yellm1);
- double yellp1;
- double yell = r_yell.val;
- double yellm1 = r_yellm1.val;
- int ell;
- result_array[0] = yellm1;
- result_array[1] = yell;
- for(ell = 1; ell < lmax; ell++) {
- yellp1 = (2*ell+1)/x * yell - yellm1;
- result_array[ell+1] = yellp1;
- yellm1 = yell;
- yell = yellp1;
- }
- return GSL_ERROR_SELECT_2(stat_0, stat_1);
- }
- }
- /*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
- #include "gsl_specfunc__eval.h"
- double gsl_sf_bessel_y0(const double x)
- {
- EVAL_RESULT(gsl_sf_bessel_y0_e(x, &result));
- }
- double gsl_sf_bessel_y1(const double x)
- {
- EVAL_RESULT(gsl_sf_bessel_y1_e(x, &result));
- }
- double gsl_sf_bessel_y2(const double x)
- {
- EVAL_RESULT(gsl_sf_bessel_y2_e(x, &result));
- }
- double gsl_sf_bessel_yl(const int l, const double x)
- {
- EVAL_RESULT(gsl_sf_bessel_yl_e(l, x, &result));
- }
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