/* 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));

}