| /* specfunc/bessel_j.c |
| * |
| * Copyright (C) 1996,1997,1998,1999,2000,2001,2002,2003 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_trig.h> |
| #include <gsl/gsl_sf_bessel.h> |
| |
| #include "error.h" |
| |
| #include "bessel.h" |
| #include "bessel_olver.h" |
| |
| /*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/ |
| |
| int gsl_sf_bessel_j0_e(const double x, gsl_sf_result * result) |
| { |
| double ax = fabs(x); |
| |
| /* CHECK_POINTER(result) */ |
| |
| if(ax < 0.5) { |
| const double y = x*x; |
| 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 c6 = 1.0/6227020800.0; |
| result->val = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*(c5 + y*c6))))); |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_SUCCESS; |
| } |
| else { |
| gsl_sf_result sin_result; |
| const int stat = gsl_sf_sin_e(x, &sin_result); |
| result->val = sin_result.val/x; |
| result->err = fabs(sin_result.err/x); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return stat; |
| } |
| } |
| |
| |
| int gsl_sf_bessel_j1_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 < 3.1*GSL_DBL_MIN) { |
| UNDERFLOW_ERROR(result); |
| } |
| else if(ax < 0.25) { |
| 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 = x/3.0 * sum; |
| result->err = 2.0 * 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 cos_x = cos_result.val; |
| const double sin_x = sin_result.val; |
| result->val = (sin_x/x - cos_x)/x; |
| result->err = (fabs(sin_result.err/x) + fabs(cos_result.err))/fabs(x); |
| result->err += 2.0 * GSL_DBL_EPSILON * (fabs(sin_x/(x*x)) + fabs(cos_x/x)); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_ERROR_SELECT_2(stat_cos, stat_sin); |
| } |
| } |
| |
| |
| int gsl_sf_bessel_j2_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 < 1.3) { |
| 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; |
| const double c6 = 1.0/105859353600.0; |
| const double c7 = -1.0/28158588057600.0; |
| const double c8 = 1.0/9461285587353600.0; |
| const double c9 = -1.0/3916972233164390400.0; |
| const double sum = 1.0+y*(c1+y*(c2+y*(c3+y*(c4+y*(c5+y*(c6+y*(c7+y*(c8+y*c9)))))))); |
| result->val = y/15.0 * sum; |
| result->err = 2.0 * 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 cos_x = cos_result.val; |
| const double sin_x = sin_result.val; |
| const double f = (3.0/(x*x) - 1.0); |
| result->val = (f * sin_x - 3.0*cos_x/x)/x; |
| result->err = fabs(f * sin_result.err/x) + fabs((3.0*cos_result.err/x)/x); |
| result->err += 2.0 * GSL_DBL_EPSILON * (fabs(f*sin_x/x) + 3.0*fabs(cos_x/(x*x))); |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return GSL_ERROR_SELECT_2(stat_cos, stat_sin); |
| } |
| } |
| |
| |
| int |
| gsl_sf_bessel_jl_e(const int l, const double x, gsl_sf_result * result) |
| { |
| if(l < 0 || x < 0.0) { |
| DOMAIN_ERROR(result); |
| } |
| else if(x == 0.0) { |
| result->val = ( l > 0 ? 0.0 : 1.0 ); |
| result->err = 0.0; |
| return GSL_SUCCESS; |
| } |
| else if(l == 0) { |
| return gsl_sf_bessel_j0_e(x, result); |
| } |
| else if(l == 1) { |
| return gsl_sf_bessel_j1_e(x, result); |
| } |
| else if(l == 2) { |
| return gsl_sf_bessel_j2_e(x, result); |
| } |
| else if(x*x < 10.0*(l+0.5)/M_E) { |
| gsl_sf_result b; |
| int status = gsl_sf_bessel_IJ_taylor_e(l+0.5, x, -1, 50, GSL_DBL_EPSILON, &b); |
| double pre = sqrt((0.5*M_PI)/x); |
| result->val = pre * b.val; |
| result->err = pre * b.err; |
| result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val); |
| return status; |
| } |
| else if(GSL_ROOT4_DBL_EPSILON * x > (l*l + l + 1.0)) { |
| gsl_sf_result b; |
| int status = gsl_sf_bessel_Jnu_asympx_e(l + 0.5, x, &b); |
| double pre = sqrt((0.5*M_PI)/x); |
| result->val = pre * b.val; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val) + pre * b.err; |
| return status; |
| } |
| else if(l > 1.0/GSL_ROOT6_DBL_EPSILON) { |
| gsl_sf_result b; |
| int status = gsl_sf_bessel_Jnu_asymp_Olver_e(l + 0.5, x, &b); |
| double pre = sqrt((0.5*M_PI)/x); |
| result->val = pre * b.val; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val) + pre * b.err; |
| return status; |
| } |
| else if(x > 1000.0 && x > 100.0*l*l) |
| { |
| /* We need this to avoid feeding large x to CF1; note that |
| * due to the above check, we know that n <= 50. |
| */ |
| gsl_sf_result b; |
| int status = gsl_sf_bessel_Jnu_asympx_e(l + 0.5, x, &b); |
| double pre = sqrt((0.5*M_PI)/x); |
| result->val = pre * b.val; |
| result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val) + pre * b.err; |
| return status; |
| } |
| else { |
| double sgn; |
| double ratio; |
| int stat_CF1 = gsl_sf_bessel_J_CF1(l+0.5, x, &ratio, &sgn); |
| double jellp1 = GSL_SQRT_DBL_EPSILON * ratio; |
| double jell = GSL_SQRT_DBL_EPSILON; |
| double jellm1; |
| int ell; |
| for(ell = l; ell > 0; ell--) { |
| jellm1 = -jellp1 + (2*ell + 1)/x * jell; |
| jellp1 = jell; |
| jell = jellm1; |
| } |
| |
| if(fabs(jell) > fabs(jellp1)) { |
| gsl_sf_result j0_result; |
| int stat_j0 = gsl_sf_bessel_j0_e(x, &j0_result); |
| double pre = GSL_SQRT_DBL_EPSILON / jell; |
| result->val = j0_result.val * pre; |
| result->err = j0_result.err * fabs(pre); |
| result->err += 2.0 * GSL_DBL_EPSILON * (0.5*l + 1.0) * fabs(result->val); |
| return GSL_ERROR_SELECT_2(stat_j0, stat_CF1); |
| } |
| else { |
| gsl_sf_result j1_result; |
| int stat_j1 = gsl_sf_bessel_j1_e(x, &j1_result); |
| double pre = GSL_SQRT_DBL_EPSILON / jellp1; |
| result->val = j1_result.val * pre; |
| result->err = j1_result.err * fabs(pre); |
| result->err += 2.0 * GSL_DBL_EPSILON * (0.5*l + 1.0) * fabs(result->val); |
| return GSL_ERROR_SELECT_2(stat_j1, stat_CF1); |
| } |
| } |
| } |
| |
| |
| int |
| gsl_sf_bessel_jl_array(const int lmax, const double x, double * result_array) |
| { |
| /* CHECK_POINTER(result_array) */ |
| |
| if(lmax < 0 || x < 0.0) { |
| int j; |
| for(j=0; j<=lmax; j++) result_array[j] = 0.0; |
| GSL_ERROR ("error", GSL_EDOM); |
| } |
| else if(x == 0.0) { |
| int j; |
| for(j=1; j<=lmax; j++) result_array[j] = 0.0; |
| result_array[0] = 1.0; |
| return GSL_SUCCESS; |
| } |
| else { |
| gsl_sf_result r_jellp1; |
| gsl_sf_result r_jell; |
| int stat_0 = gsl_sf_bessel_jl_e(lmax+1, x, &r_jellp1); |
| int stat_1 = gsl_sf_bessel_jl_e(lmax, x, &r_jell); |
| double jellp1 = r_jellp1.val; |
| double jell = r_jell.val; |
| double jellm1; |
| int ell; |
| |
| result_array[lmax] = jell; |
| for(ell = lmax; ell >= 1; ell--) { |
| jellm1 = -jellp1 + (2*ell + 1)/x * jell; |
| jellp1 = jell; |
| jell = jellm1; |
| result_array[ell-1] = jellm1; |
| } |
| |
| return GSL_ERROR_SELECT_2(stat_0, stat_1); |
| } |
| } |
| |
| |
| int gsl_sf_bessel_jl_steed_array(const int lmax, const double x, double * jl_x) |
| { |
| /* CHECK_POINTER(jl_x) */ |
| |
| if(lmax < 0 || x < 0.0) { |
| int j; |
| for(j=0; j<=lmax; j++) jl_x[j] = 0.0; |
| GSL_ERROR ("error", GSL_EDOM); |
| } |
| else if(x == 0.0) { |
| int j; |
| for(j=1; j<=lmax; j++) jl_x[j] = 0.0; |
| jl_x[0] = 1.0; |
| return GSL_SUCCESS; |
| } |
| else if(x < 2.0*GSL_ROOT4_DBL_EPSILON) { |
| /* first two terms of Taylor series */ |
| double inv_fact = 1.0; /* 1/(1 3 5 ... (2l+1)) */ |
| double x_l = 1.0; /* x^l */ |
| int l; |
| for(l=0; l<=lmax; l++) { |
| jl_x[l] = x_l * inv_fact; |
| jl_x[l] *= 1.0 - 0.5*x*x/(2.0*l+3.0); |
| inv_fact /= 2.0*l+3.0; |
| x_l *= x; |
| } |
| return GSL_SUCCESS; |
| } |
| else { |
| /* Steed/Barnett algorithm [Comp. Phys. Comm. 21, 297 (1981)] */ |
| double x_inv = 1.0/x; |
| double W = 2.0*x_inv; |
| double F = 1.0; |
| double FP = (lmax+1.0) * x_inv; |
| double B = 2.0*FP + x_inv; |
| double end = B + 20000.0*W; |
| double D = 1.0/B; |
| double del = -D; |
| |
| FP += del; |
| |
| /* continued fraction */ |
| do { |
| B += W; |
| D = 1.0/(B-D); |
| del *= (B*D - 1.); |
| FP += del; |
| if(D < 0.0) F = -F; |
| if(B > end) { |
| GSL_ERROR ("error", GSL_EMAXITER); |
| } |
| } |
| while(fabs(del) >= fabs(FP) * GSL_DBL_EPSILON); |
| |
| FP *= F; |
| |
| if(lmax > 0) { |
| /* downward recursion */ |
| double XP2 = FP; |
| double PL = lmax * x_inv; |
| int L = lmax; |
| int LP; |
| jl_x[lmax] = F; |
| for(LP = 1; LP<=lmax; LP++) { |
| jl_x[L-1] = PL * jl_x[L] + XP2; |
| FP = PL*jl_x[L-1] - jl_x[L]; |
| XP2 = FP; |
| PL -= x_inv; |
| --L; |
| } |
| F = jl_x[0]; |
| } |
| |
| /* normalization */ |
| W = x_inv / hypot(FP, F); |
| jl_x[0] = W*F; |
| if(lmax > 0) { |
| int L; |
| for(L=1; L<=lmax; L++) { |
| jl_x[L] *= W; |
| } |
| } |
| |
| return GSL_SUCCESS; |
| } |
| } |
| |
| |
| /*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/ |
| |
| #include "eval.h" |
| |
| double gsl_sf_bessel_j0(const double x) |
| { |
| EVAL_RESULT(gsl_sf_bessel_j0_e(x, &result)); |
| } |
| |
| double gsl_sf_bessel_j1(const double x) |
| { |
| EVAL_RESULT(gsl_sf_bessel_j1_e(x, &result)); |
| } |
| |
| double gsl_sf_bessel_j2(const double x) |
| { |
| EVAL_RESULT(gsl_sf_bessel_j2_e(x, &result)); |
| } |
| |
| double gsl_sf_bessel_jl(const int l, const double x) |
| { |
| EVAL_RESULT(gsl_sf_bessel_jl_e(l, x, &result)); |
| } |
| |