| /* 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 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_pow_int.h> |
| #include <gsl/gsl_sf_gamma.h> |
| #include <gsl/gsl_sf_trig.h> |
| #include <gsl/gsl_sf_bessel.h> |
| |
| #include "error.h" |
| |
| #include "bessel.h" |
| #include "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 "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)); |
| } |
| |
| |