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/* specfunc/bessel_Knu.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_exp.h>
#include <gsl/gsl_sf_gamma.h>
#include <gsl/gsl_sf_bessel.h>
#include "error.h"
#include "bessel.h"
#include "bessel_temme.h"
/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
int
gsl_sf_bessel_Knu_scaled_e(const double nu, const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= 0.0 || nu < 0.0) {
DOMAIN_ERROR(result);
}
else {
gsl_sf_result_e10 result_e10;
int status = gsl_sf_bessel_Knu_scaled_e10_e(nu, x, &result_e10);
int status2 = gsl_sf_result_smash_e(&result_e10, result);
return GSL_ERROR_SELECT_2(status, status2);
}
}
int
gsl_sf_bessel_Knu_scaled_e10_e(const double nu, const double x, gsl_sf_result_e10 * result)
{
/* CHECK_POINTER(result) */
if(x <= 0.0 || nu < 0.0) {
DOMAIN_ERROR_E10(result);
}
else {
int N = (int)(nu + 0.5);
double mu = nu - N; /* -1/2 <= mu <= 1/2 */
double K_mu, K_mup1, Kp_mu;
double K_nu, K_nup1, K_num1;
int n, e10 = 0;
if(x < 2.0) {
gsl_sf_bessel_K_scaled_temme(mu, x, &K_mu, &K_mup1, &Kp_mu);
}
else {
gsl_sf_bessel_K_scaled_steed_temme_CF2(mu, x, &K_mu, &K_mup1, &Kp_mu);
}
/* recurse forward to obtain K_num1, K_nu */
K_nu = K_mu;
K_nup1 = K_mup1;
for(n=0; n<N; n++) {
K_num1 = K_nu;
K_nu = K_nup1;
/* rescale the recurrence to avoid overflow */
if (fabs(K_nu) > GSL_SQRT_DBL_MAX) {
double p = floor(log(fabs(K_nu))/M_LN10);
double factor = pow(10.0, p);
K_num1 /= factor;
K_nu /= factor;
e10 += p;
};
K_nup1 = 2.0*(mu+n+1)/x * K_nu + K_num1;
}
result->val = K_nu;
result->err = 2.0 * GSL_DBL_EPSILON * (N + 4.0) * fabs(result->val);
result->e10 = e10;
return GSL_SUCCESS;
}
}
int
gsl_sf_bessel_Knu_e(const double nu, const double x, gsl_sf_result * result)
{
gsl_sf_result b;
int stat_K = gsl_sf_bessel_Knu_scaled_e(nu, x, &b);
int stat_e = gsl_sf_exp_mult_err_e(-x, 0.0, b.val, b.err, result);
return GSL_ERROR_SELECT_2(stat_e, stat_K);
}
int
gsl_sf_bessel_lnKnu_e(const double nu, const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= 0.0 || nu < 0.0) {
DOMAIN_ERROR(result);
}
else if(nu == 0.0) {
gsl_sf_result K_scaled;
/* This cannot underflow, and
* it will not throw GSL_EDOM
* since that is already checked.
*/
gsl_sf_bessel_K0_scaled_e(x, &K_scaled);
result->val = -x + log(fabs(K_scaled.val));
result->err = GSL_DBL_EPSILON * fabs(x) + fabs(K_scaled.err/K_scaled.val);
result->err += GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
else if(x < 2.0 && nu > 1.0) {
/* Make use of the inequality
* Knu(x) <= 1/2 (2/x)^nu Gamma(nu),
* which follows from the integral representation
* [Abramowitz+Stegun, 9.6.23 (2)]. With this
* we decide whether or not there is an overflow
* problem because x is small.
*/
double ln_bound;
gsl_sf_result lg_nu;
gsl_sf_lngamma_e(nu, &lg_nu);
ln_bound = -M_LN2 - nu*log(0.5*x) + lg_nu.val;
if(ln_bound > GSL_LOG_DBL_MAX - 20.0) {
/* x must be very small or nu very large (or both).
*/
double xi = 0.25*x*x;
double sum = 1.0 - xi/(nu-1.0);
if(nu > 2.0) sum += (xi/(nu-1.0)) * (xi/(nu-2.0));
result->val = ln_bound + log(sum);
result->err = lg_nu.err;
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
/* can drop-through here */
}
{
/* We passed the above tests, so no problem.
* Evaluate as usual. Note the possible drop-through
* in the above code!
*/
gsl_sf_result_e10 K_scaled;
int status = gsl_sf_bessel_Knu_scaled_e10_e(nu, x, &K_scaled);
result->val = -x + log(fabs(K_scaled.val)) + K_scaled.e10 * M_LN10;
result->err = GSL_DBL_EPSILON * fabs(x) + fabs(K_scaled.err/K_scaled.val);
result->err += GSL_DBL_EPSILON * fabs(result->val);
return status;
}
}
/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
#include "eval.h"
double gsl_sf_bessel_Knu_scaled(const double nu, const double x)
{
EVAL_RESULT(gsl_sf_bessel_Knu_scaled_e(nu, x, &result));
}
double gsl_sf_bessel_Knu(const double nu, const double x)
{
EVAL_RESULT(gsl_sf_bessel_Knu_e(nu, x, &result));
}
double gsl_sf_bessel_lnKnu(const double nu, const double x)
{
EVAL_RESULT(gsl_sf_bessel_lnKnu_e(nu, x, &result));
}
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