/* 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 = 1.0/3486356.0;

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));

}