/* 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 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 <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 {
result->val = sin(x) / x;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
}
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 {
const double cos_x = cos(x);
const double sin_x = sin(x);
result->val = (sin_x/x - cos_x)/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_SUCCESS;
}
}
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 {
/* bug #45730: switch from gsl_sf_{cos,sin} to cos/sin to fix large inputs */
#if 0
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);
#else
const double cos_x = cos(x);
const double sin_x = sin(x);
const double f = (3.0/(x*x) - 1.0);
result->val = (f * sin_x - 3.0*cos_x/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);*/
return GSL_SUCCESS;
#endif
}
}
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 > l*l)
{
/* We need this path to avoid feeding large x to CF1 below; */
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;
/* The CF1 call will hit 10000 iterations for x > 10000 + l */
int stat_CF1 = gsl_sf_bessel_J_CF1(l+0.5, x, &ratio, &sgn);
const double BESSEL_J_SMALL = GSL_DBL_MIN / GSL_DBL_EPSILON;
double jellp1 = BESSEL_J_SMALL * ratio;
double jell = BESSEL_J_SMALL;
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 = BESSEL_J_SMALL / jell;
result->val = j0_result.val * pre;
result->err = j0_result.err * fabs(pre);
result->err += 4.0 * 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 = BESSEL_J_SMALL / jellp1;
result->val = j1_result.val * pre;
result->err = j1_result.err * fabs(pre);
result->err += 4.0 * 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));
}