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/* specfunc/bessel_k.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_pow_int.h>
#include <gsl/gsl_sf_gamma.h>
#include <gsl/gsl_sf_bessel.h>
#include "error.h"
#include "check.h"
#include "bessel.h"
/*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/
/* [Abramowitz+Stegun, 10.2.4 + 10.2.6]
* with lmax=15, precision ~ 15D for x < 3
*
* assumes l >= 1
*/
static int bessel_kl_scaled_small_x(int l, const double x, gsl_sf_result * result)
{
gsl_sf_result num_fact;
double den = gsl_sf_pow_int(x, l+1);
int stat_df = gsl_sf_doublefact_e((unsigned int) (2*l-1), &num_fact);
if(stat_df != GSL_SUCCESS || den == 0.0) {
OVERFLOW_ERROR(result);
}
else {
const int lmax = 50;
gsl_sf_result ipos_term;
double ineg_term;
double sgn = (GSL_IS_ODD(l) ? -1.0 : 1.0);
double ex = exp(x);
double t = 0.5*x*x;
double sum = 1.0;
double t_coeff = 1.0;
double t_power = 1.0;
double delta;
int stat_il;
int i;
for(i=1; i<lmax; i++) {
t_coeff /= i*(2*(i-l) - 1);
t_power *= t;
delta = t_power*t_coeff;
sum += delta;
if(fabs(delta/sum) < GSL_DBL_EPSILON) break;
}
stat_il = gsl_sf_bessel_il_scaled_e(l, x, &ipos_term);
ineg_term = sgn * num_fact.val/den * sum;
result->val = -sgn * 0.5*M_PI * (ex*ipos_term.val - ineg_term);
result->val *= ex;
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return stat_il;
}
}
/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
int gsl_sf_bessel_k0_scaled_e(const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= 0.0) {
DOMAIN_ERROR(result);
}
else {
result->val = M_PI/(2.0*x);
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
CHECK_UNDERFLOW(result);
return GSL_SUCCESS;
}
}
int gsl_sf_bessel_k1_scaled_e(const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= 0.0) {
DOMAIN_ERROR(result);
}
else if(x < (M_SQRTPI+1.0)/(M_SQRT2*GSL_SQRT_DBL_MAX)) {
OVERFLOW_ERROR(result);
}
else {
result->val = M_PI/(2.0*x) * (1.0 + 1.0/x);
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
CHECK_UNDERFLOW(result);
return GSL_SUCCESS;
}
}
int gsl_sf_bessel_k2_scaled_e(const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= 0.0) {
DOMAIN_ERROR(result);
}
else if(x < 2.0/GSL_ROOT3_DBL_MAX) {
OVERFLOW_ERROR(result);
}
else {
result->val = M_PI/(2.0*x) * (1.0 + 3.0/x * (1.0 + 1.0/x));
result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
CHECK_UNDERFLOW(result);
return GSL_SUCCESS;
}
}
int gsl_sf_bessel_kl_scaled_e(int l, const double x, gsl_sf_result * result)
{
if(l < 0 || x <= 0.0) {
DOMAIN_ERROR(result);
}
else if(l == 0) {
return gsl_sf_bessel_k0_scaled_e(x, result);
}
else if(l == 1) {
return gsl_sf_bessel_k1_scaled_e(x, result);
}
else if(l == 2) {
return gsl_sf_bessel_k2_scaled_e(x, result);
}
else if(x < 3.0) {
return bessel_kl_scaled_small_x(l, x, result);
}
else if(GSL_ROOT3_DBL_EPSILON * x > (l*l + l + 1)) {
int status = gsl_sf_bessel_Knu_scaled_asympx_e(l + 0.5, x, result);
double pre = sqrt((0.5*M_PI)/x);
result->val *= pre;
result->err *= pre;
return status;
}
else if(GSL_MIN(0.29/(l*l+1.0), 0.5/(l*l+1.0+x*x)) < GSL_ROOT3_DBL_EPSILON) {
int status = gsl_sf_bessel_Knu_scaled_asymp_unif_e(l + 0.5, x, result);
double pre = sqrt((0.5*M_PI)/x);
result->val *= pre;
result->err *= pre;
return status;
}
else {
/* recurse upward */
gsl_sf_result r_bk;
gsl_sf_result r_bkm;
int stat_1 = gsl_sf_bessel_k1_scaled_e(x, &r_bk);
int stat_0 = gsl_sf_bessel_k0_scaled_e(x, &r_bkm);
double bkp;
double bk = r_bk.val;
double bkm = r_bkm.val;
int j;
for(j=1; j<l; j++) {
bkp = (2*j+1)/x*bk + bkm;
bkm = bk;
bk = bkp;
}
result->val = bk;
result->err = fabs(bk) * (fabs(r_bk.err/r_bk.val) + fabs(r_bkm.err/r_bkm.val));
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_1, stat_0);
}
}
int
gsl_sf_bessel_kl_scaled_array(const int lmax, const double x, double * result_array)
{
if(lmax < 0 || x <= 0.0) {
GSL_ERROR("domain error", GSL_EDOM);
} else if (lmax == 0) {
gsl_sf_result result;
int stat = gsl_sf_bessel_k0_scaled_e(x, &result);
result_array[0] = result.val;
return stat;
} else {
int ell;
double kellp1, kell, kellm1;
gsl_sf_result r_kell;
gsl_sf_result r_kellm1;
gsl_sf_bessel_k1_scaled_e(x, &r_kell);
gsl_sf_bessel_k0_scaled_e(x, &r_kellm1);
kell = r_kell.val;
kellm1 = r_kellm1.val;
result_array[0] = kellm1;
result_array[1] = kell;
for(ell = 1; ell < lmax; ell++) {
kellp1 = (2*ell+1)/x * kell + kellm1;
result_array[ell+1] = kellp1;
kellm1 = kell;
kell = kellp1;
}
return GSL_SUCCESS;
}
}
/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
#include "eval.h"
double gsl_sf_bessel_k0_scaled(const double x)
{
EVAL_RESULT(gsl_sf_bessel_k0_scaled_e(x, &result));
}
double gsl_sf_bessel_k1_scaled(const double x)
{
EVAL_RESULT(gsl_sf_bessel_k1_scaled_e(x, &result));
}
double gsl_sf_bessel_k2_scaled(const double x)
{
EVAL_RESULT(gsl_sf_bessel_k2_scaled_e(x, &result));
}
double gsl_sf_bessel_kl_scaled(const int l, const double x)
{
EVAL_RESULT(gsl_sf_bessel_kl_scaled_e(l, x, &result));
}
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