| /* specfunc/bessel_i.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_bessel.h> |
| |
| #include "error.h" |
| |
| #include "bessel.h" |
| |
| |
| /* i_{l+1}/i_l |
| */ |
| static |
| int |
| bessel_il_CF1(const int l, const double x, const double threshold, double * ratio) |
| { |
| const int kmax = 2000; |
| double tk = 1.0; |
| double sum = 1.0; |
| double rhok = 0.0; |
| int k; |
| |
| for(k=1; k<=kmax; k++) { |
| double ak = (x/(2.0*l+1.0+2.0*k)) * (x/(2.0*l+3.0+2.0*k)); |
| rhok = -ak*(1.0 + rhok)/(1.0 + ak*(1.0 + rhok)); |
| tk *= rhok; |
| sum += tk; |
| if(fabs(tk/sum) < threshold) break; |
| } |
| |
| *ratio = x/(2.0*l+3.0) * sum; |
| |
| if(k == kmax) |
| GSL_ERROR ("error", GSL_EMAXITER); |
| else |
| return GSL_SUCCESS; |
| } |
| |
| |
| /*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/ |
| |
| int gsl_sf_bessel_i0_scaled_e(const double x, gsl_sf_result * result) |
| { |
| double ax = fabs(x); |
| |
| /* CHECK_POINTER(result) */ |
| |
| if(x == 0.0) { |
| result->val = 1.0; |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else if(ax < 0.2) { |
| const double eax = exp(-ax); |
| const double y = ax*ax; |
| const double c1 = 1.0/6.0; |
| const double c2 = 1.0/120.0; |
| const double c3 = 1.0/5040.0; |
| const double c4 = 1.0/362880.0; |
| const double c5 = 1.0/39916800.0; |
| const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*c5)))); |
| result->val = eax * sum; |
| result->err = 2.0 * GSL_DBL_EPSILON * result->val; |
| } |
| else if(ax < -0.5*GSL_LOG_DBL_EPSILON) { |
| result->val = (1.0 - exp(-2.0*ax))/(2.0*ax); |
| result->err = 2.0 * GSL_DBL_EPSILON * result->val; |
| } |
| else { |
| result->val = 1.0/(2.0*ax); |
| result->err = 2.0 * GSL_DBL_EPSILON * result->val; |
| } |
| return GSL_SUCCESS; |
| } |
| |
| |
| int gsl_sf_bessel_i1_scaled_e(const double x, gsl_sf_result * result) |
| { |
| double ax = fabs(x); |
| |
| /* CHECK_POINTER(result) */ |
| |
| if(x == 0.0) { |
| result->val = 0.0; |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else if(ax < 3.0*GSL_DBL_MIN) { |
| UNDERFLOW_ERROR(result); |
| } |
| else if(ax < 0.25) { |
| const double eax = exp(-ax); |
| const double y = x*x; |
| const double c1 = 1.0/10.0; |
| const double c2 = 1.0/280.0; |
| const double c3 = 1.0/15120.0; |
| const double c4 = 1.0/1330560.0; |
| const double c5 = 1.0/172972800.0; |
| const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*c5)))); |
| result->val = eax * x/3.0 * sum; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| double ex = exp(-2.0*ax); |
| result->val = 0.5 * (ax*(1.0+ex) - (1.0-ex)) / (ax*ax); |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| if(x < 0.0) result->val = -result->val; |
| return GSL_SUCCESS; |
| } |
| } |
| |
| |
| int gsl_sf_bessel_i2_scaled_e(const double x, gsl_sf_result * result) |
| { |
| double ax = fabs(x); |
| |
| /* CHECK_POINTER(result) */ |
| |
| if(x == 0.0) { |
| result->val = 0.0; |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else if(ax < 4.0*GSL_SQRT_DBL_MIN) { |
| UNDERFLOW_ERROR(result); |
| } |
| else if(ax < 0.25) { |
| const double y = x*x; |
| const double c1 = 1.0/14.0; |
| const double c2 = 1.0/504.0; |
| const double c3 = 1.0/33264.0; |
| const double c4 = 1.0/3459456.0; |
| const double c5 = 1.0/518918400.0; |
| const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*c5)))); |
| const double pre = exp(-ax) * x*x/15.0; |
| result->val = pre * sum; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| double ex = exp(-2.0*ax); |
| double x2 = x*x; |
| result->val = 0.5 * ((3.0+x2)*(1.0-ex) - 3.0*ax*(1.0+ex))/(ax*ax*ax); |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| } |
| |
| |
| int gsl_sf_bessel_il_scaled_e(const int l, double x, gsl_sf_result * result) |
| { |
| double sgn = 1.0; |
| double ax = fabs(x); |
| |
| if(x < 0.0) { |
| /* i_l(-x) = (-1)^l i_l(x) */ |
| sgn = ( GSL_IS_ODD(l) ? -1.0 : 1.0 ); |
| x = -x; |
| } |
| |
| if(l < 0) { |
| DOMAIN_ERROR(result); |
| } |
| else if(x == 0.0) { |
| result->val = ( l == 0 ? 1.0 : 0.0 ); |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else if(l == 0) { |
| gsl_sf_result il; |
| int stat_il = gsl_sf_bessel_i0_scaled_e(x, &il); |
| result->val = sgn * il.val; |
| result->err = il.err; |
| return stat_il; |
| } |
| else if(l == 1) { |
| gsl_sf_result il; |
| int stat_il = gsl_sf_bessel_i1_scaled_e(x, &il); |
| result->val = sgn * il.val; |
| result->err = il.err; |
| return stat_il; |
| } |
| else if(l == 2) { |
| gsl_sf_result il; |
| int stat_il = gsl_sf_bessel_i2_scaled_e(x, &il); |
| result->val = sgn * il.val; |
| result->err = il.err; |
| return stat_il; |
| } |
| else if(x*x < 10.0*(l+1.5)/M_E) { |
| gsl_sf_result b; |
| int stat = gsl_sf_bessel_IJ_taylor_e(l+0.5, x, 1, 50, GSL_DBL_EPSILON, &b); |
| double pre = exp(-ax) * sqrt((0.5*M_PI)/x); |
| result->val = sgn * pre * b.val; |
| result->err = pre * b.err; |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return stat; |
| } |
| else if(l < 150) { |
| gsl_sf_result i0_scaled; |
| int stat_i0 = gsl_sf_bessel_i0_scaled_e(ax, &i0_scaled); |
| double rat; |
| int stat_CF1 = bessel_il_CF1(l, ax, GSL_DBL_EPSILON, &rat); |
| double iellp1 = rat * GSL_SQRT_DBL_MIN; |
| double iell = GSL_SQRT_DBL_MIN; |
| double iellm1; |
| int ell; |
| for(ell = l; ell >= 1; ell--) { |
| iellm1 = iellp1 + (2*ell + 1)/x * iell; |
| iellp1 = iell; |
| iell = iellm1; |
| } |
| result->val = sgn * i0_scaled.val * (GSL_SQRT_DBL_MIN / iell); |
| result->err = i0_scaled.err * (GSL_SQRT_DBL_MIN / iell); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_ERROR_SELECT_2(stat_i0, stat_CF1); |
| } |
| else if(GSL_MIN(0.29/(l*l+1.0), 0.5/(l*l+1.0+x*x)) < 0.5*GSL_ROOT3_DBL_EPSILON) { |
| int status = gsl_sf_bessel_Inu_scaled_asymp_unif_e(l + 0.5, x, result); |
| double pre = sqrt((0.5*M_PI)/x); |
| result->val *= sgn * pre; |
| result->err *= pre; |
| return status; |
| } |
| else { |
| /* recurse down from safe values */ |
| double rt_term = sqrt((0.5*M_PI)/x); |
| const int LMAX = 2 + (int) (1.2 / GSL_ROOT6_DBL_EPSILON); |
| gsl_sf_result r_iellp1; |
| gsl_sf_result r_iell; |
| int stat_a1 = gsl_sf_bessel_Inu_scaled_asymp_unif_e(LMAX + 1 + 0.5, x, &r_iellp1); |
| int stat_a2 = gsl_sf_bessel_Inu_scaled_asymp_unif_e(LMAX + 0.5, x, &r_iell); |
| double iellp1 = r_iellp1.val; |
| double iell = r_iell.val; |
| double iellm1 = 0.0; |
| int ell; |
| iellp1 *= rt_term; |
| iell *= rt_term; |
| for(ell = LMAX; ell >= l+1; ell--) { |
| iellm1 = iellp1 + (2*ell + 1)/x * iell; |
| iellp1 = iell; |
| iell = iellm1; |
| } |
| result->val = sgn * iellm1; |
| result->err = fabs(result->val)*(GSL_DBL_EPSILON + fabs(r_iellp1.err/r_iellp1.val) + fabs(r_iell.err/r_iell.val)); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| |
| return GSL_ERROR_SELECT_2(stat_a1, stat_a2); |
| } |
| } |
| |
| |
| int gsl_sf_bessel_il_scaled_array(const int lmax, const double x, double * result_array) |
| { |
| if(x == 0.0) { |
| int ell; |
| result_array[0] = 1.0; |
| for (ell = lmax; ell >= 1; ell--) { |
| result_array[ell] = 0.0; |
| }; |
| return GSL_SUCCESS; |
| } else { |
| int ell; |
| gsl_sf_result r_iellp1; |
| gsl_sf_result r_iell; |
| int stat_0 = gsl_sf_bessel_il_scaled_e(lmax+1, x, &r_iellp1); |
| int stat_1 = gsl_sf_bessel_il_scaled_e(lmax, x, &r_iell); |
| double iellp1 = r_iellp1.val; |
| double iell = r_iell.val; |
| double iellm1; |
| result_array[lmax] = iell; |
| for(ell = lmax; ell >= 1; ell--) { |
| iellm1 = iellp1 + (2*ell + 1)/x * iell; |
| iellp1 = iell; |
| iell = iellm1; |
| result_array[ell-1] = iellm1; |
| } |
| return GSL_ERROR_SELECT_2(stat_0, stat_1); |
| } |
| } |
| |
| |
| /*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/ |
| |
| #include "eval.h" |
| |
| double gsl_sf_bessel_i0_scaled(const double x) |
| { |
| EVAL_RESULT(gsl_sf_bessel_i0_scaled_e(x, &result)); |
| } |
| |
| double gsl_sf_bessel_i1_scaled(const double x) |
| { |
| EVAL_RESULT(gsl_sf_bessel_i1_scaled_e(x, &result)); |
| } |
| |
| double gsl_sf_bessel_i2_scaled(const double x) |
| { |
| EVAL_RESULT(gsl_sf_bessel_i2_scaled_e(x, &result)); |
| } |
| |
| double gsl_sf_bessel_il_scaled(const int l, const double x) |
| { |
| EVAL_RESULT(gsl_sf_bessel_il_scaled_e(l, x, &result)); |
| } |
| |
