/* 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 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_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));
}