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/* specfunc/bessel_In.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_bessel.h>
#include "error.h"
#include "bessel.h"
/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
int
gsl_sf_bessel_In_scaled_e(int n, const double x, gsl_sf_result * result)
{
const double ax = fabs(x);
n = abs(n); /* I(-n, z) = I(n, z) */
/* CHECK_POINTER(result) */
if(n == 0) {
return gsl_sf_bessel_I0_scaled_e(x, result);
}
else if(n == 1) {
return gsl_sf_bessel_I1_scaled_e(x, result);
}
else if(x == 0.0) {
result->val = 0.0;
result->err = 0.0;
return GSL_SUCCESS;
}
else if(x*x < 10.0*(n+1.0)/M_E) {
gsl_sf_result t;
double ex = exp(-ax);
int stat_In = gsl_sf_bessel_IJ_taylor_e((double)n, ax, 1, 50, GSL_DBL_EPSILON, &t);
result->val = t.val * ex;
result->err = t.err * ex;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
if(x < 0.0 && GSL_IS_ODD(n)) result->val = -result->val;
return stat_In;
}
else if(n < 150 && ax < 1e7) {
gsl_sf_result I0_scaled;
int stat_I0 = gsl_sf_bessel_I0_scaled_e(ax, &I0_scaled);
double rat;
int stat_CF1 = gsl_sf_bessel_I_CF1_ser((double)n, ax, &rat);
double Ikp1 = rat * GSL_SQRT_DBL_MIN;
double Ik = GSL_SQRT_DBL_MIN;
double Ikm1;
int k;
for(k=n; k >= 1; k--) {
Ikm1 = Ikp1 + 2.0*k/ax * Ik;
Ikp1 = Ik;
Ik = Ikm1;
}
result->val = I0_scaled.val * (GSL_SQRT_DBL_MIN / Ik);
result->err = I0_scaled.err * (GSL_SQRT_DBL_MIN / Ik);
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
if(x < 0.0 && GSL_IS_ODD(n)) result->val = -result->val;
return GSL_ERROR_SELECT_2(stat_I0, stat_CF1);
}
else if( GSL_MIN( 0.29/(n*n), 0.5/(n*n + x*x) ) < 0.5*GSL_ROOT3_DBL_EPSILON) {
int stat_as = gsl_sf_bessel_Inu_scaled_asymp_unif_e((double)n, ax, result);
if(x < 0.0 && GSL_IS_ODD(n)) result->val = -result->val;
return stat_as;
}
else {
const int nhi = 2 + (int) (1.2 / GSL_ROOT6_DBL_EPSILON);
gsl_sf_result r_Ikp1;
gsl_sf_result r_Ik;
int stat_a1 = gsl_sf_bessel_Inu_scaled_asymp_unif_e(nhi+1.0, ax, &r_Ikp1);
int stat_a2 = gsl_sf_bessel_Inu_scaled_asymp_unif_e((double)nhi, ax, &r_Ik);
double Ikp1 = r_Ikp1.val;
double Ik = r_Ik.val;
double Ikm1;
int k;
for(k=nhi; k > n; k--) {
Ikm1 = Ikp1 + 2.0*k/ax * Ik;
Ikp1 = Ik;
Ik = Ikm1;
}
result->val = Ik;
result->err = Ik * (r_Ikp1.err/r_Ikp1.val + r_Ik.err/r_Ik.val);
if(x < 0.0 && GSL_IS_ODD(n)) result->val = -result->val;
return GSL_ERROR_SELECT_2(stat_a1, stat_a2);
}
}
int
gsl_sf_bessel_In_scaled_array(const int nmin, const int nmax, const double x, double * result_array)
{
/* CHECK_POINTER(result_array) */
if(nmax < nmin || nmin < 0) {
int j;
for(j=0; j<=nmax-nmin; j++) result_array[j] = 0.0;
GSL_ERROR ("domain error", GSL_EDOM);
}
else if(x == 0.0) {
int j;
for(j=0; j<=nmax-nmin; j++) result_array[j] = 0.0;
if(nmin == 0) result_array[0] = 1.0;
return GSL_SUCCESS;
}
else if(nmax == 0) {
gsl_sf_result I0_scaled;
int stat = gsl_sf_bessel_I0_scaled_e(x, &I0_scaled);
result_array[0] = I0_scaled.val;
return stat;
}
else {
const double ax = fabs(x);
const double two_over_x = 2.0/ax;
/* starting values */
gsl_sf_result r_Inp1;
gsl_sf_result r_In;
int stat_0 = gsl_sf_bessel_In_scaled_e(nmax+1, ax, &r_Inp1);
int stat_1 = gsl_sf_bessel_In_scaled_e(nmax, ax, &r_In);
double Inp1 = r_Inp1.val;
double In = r_In.val;
double Inm1;
int n;
for(n=nmax; n>=nmin; n--) {
result_array[n-nmin] = In;
Inm1 = Inp1 + n * two_over_x * In;
Inp1 = In;
In = Inm1;
}
/* deal with signs */
if(x < 0.0) {
for(n=nmin; n<=nmax; n++) {
if(GSL_IS_ODD(n)) result_array[n-nmin] = -result_array[n-nmin];
}
}
return GSL_ERROR_SELECT_2(stat_0, stat_1);
}
}
int
gsl_sf_bessel_In_e(const int n_in, const double x, gsl_sf_result * result)
{
const double ax = fabs(x);
const int n = abs(n_in); /* I(-n, z) = I(n, z) */
gsl_sf_result In_scaled;
const int stat_In_scaled = gsl_sf_bessel_In_scaled_e(n, ax, &In_scaled);
/* In_scaled is always less than 1,
* so this overflow check is conservative.
*/
if(ax > GSL_LOG_DBL_MAX - 1.0) {
OVERFLOW_ERROR(result);
}
else {
const double ex = exp(ax);
result->val = ex * In_scaled.val;
result->err = ex * In_scaled.err;
result->err += ax * GSL_DBL_EPSILON * fabs(result->val);
if(x < 0.0 && GSL_IS_ODD(n)) result->val = -result->val;
return stat_In_scaled;
}
}
int
gsl_sf_bessel_In_array(const int nmin, const int nmax, const double x, double * result_array)
{
double ax = fabs(x);
/* CHECK_POINTER(result_array) */
if(ax > GSL_LOG_DBL_MAX - 1.0) {
int j;
for(j=0; j<=nmax-nmin; j++) result_array[j] = 0.0; /* FIXME: should be Inf */
GSL_ERROR ("overflow", GSL_EOVRFLW);
}
else {
int j;
double eax = exp(ax);
int status = gsl_sf_bessel_In_scaled_array(nmin, nmax, x, result_array);
for(j=0; j<=nmax-nmin; j++) result_array[j] *= eax;
return status;
}
}
/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
#include "eval.h"
double gsl_sf_bessel_In_scaled(const int n, const double x)
{
EVAL_RESULT(gsl_sf_bessel_In_scaled_e(n, x, &result));
}
double gsl_sf_bessel_In(const int n, const double x)
{
EVAL_RESULT(gsl_sf_bessel_In_e(n, x, &result));
}
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