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/* specfunc/bessel_Jnu.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman, 2017 Konrad Griessinger
*
* 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_bessel.h>
#include <gsl/gsl_sf_sincos_pi.h>
#include "error.h"
#include "bessel.h"
#include "bessel_olver.h"
#include "bessel_temme.h"
/* Evaluate at large enough nu to apply asymptotic
* results and apply backward recurrence.
*/
#if 0
static
int
bessel_J_recur_asymp(const double nu, const double x,
gsl_sf_result * Jnu, gsl_sf_result * Jnup1)
{
const double nu_cut = 25.0;
int n;
int steps = ceil(nu_cut - nu) + 1;
gsl_sf_result r_Jnp1;
gsl_sf_result r_Jn;
int stat_O1 = gsl_sf_bessel_Jnu_asymp_Olver_e(nu + steps + 1.0, x, &r_Jnp1);
int stat_O2 = gsl_sf_bessel_Jnu_asymp_Olver_e(nu + steps, x, &r_Jn);
double r_fe = fabs(r_Jnp1.err/r_Jnp1.val) + fabs(r_Jn.err/r_Jn.val);
double Jnp1 = r_Jnp1.val;
double Jn = r_Jn.val;
double Jnm1;
double Jnp1_save;
for(n=steps; n>0; n--) {
Jnm1 = 2.0*(nu+n)/x * Jn - Jnp1;
Jnp1 = Jn;
Jnp1_save = Jn;
Jn = Jnm1;
}
Jnu->val = Jn;
Jnu->err = (r_fe + GSL_DBL_EPSILON * (steps + 1.0)) * fabs(Jn);
Jnup1->val = Jnp1_save;
Jnup1->err = (r_fe + GSL_DBL_EPSILON * (steps + 1.0)) * fabs(Jnp1_save);
return GSL_ERROR_SELECT_2(stat_O1, stat_O2);
}
#endif
/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
int
gsl_sf_bessel_Jnupos_e(const double nu, const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x == 0.0) {
if(nu == 0.0) {
result->val = 1.0;
result->err = 0.0;
}
else {
result->val = 0.0;
result->err = 0.0;
}
return GSL_SUCCESS;
}
else if(x*x < 10.0*(nu+1.0)) {
return gsl_sf_bessel_IJ_taylor_e(nu, x, -1, 100, GSL_DBL_EPSILON, result);
}
else if(nu > 50.0) {
return gsl_sf_bessel_Jnu_asymp_Olver_e(nu, x, result);
}
else if(x > 1000.0)
{
/* We need this to avoid feeding large x to CF1; note that
* due to the above check, we know that n <= 50. See similar
* block in bessel_Jn.c.
*/
return gsl_sf_bessel_Jnu_asympx_e(nu, x, result);
}
else {
/* -1/2 <= mu <= 1/2 */
int N = (int)(nu + 0.5);
double mu = nu - N;
/* Determine the J ratio at nu.
*/
double Jnup1_Jnu;
double sgn_Jnu;
const int stat_CF1 = gsl_sf_bessel_J_CF1(nu, x, &Jnup1_Jnu, &sgn_Jnu);
if(x < 2.0) {
/* Determine Y_mu, Y_mup1 directly and recurse forward to nu.
* Then use the CF1 information to solve for J_nu and J_nup1.
*/
gsl_sf_result Y_mu, Y_mup1;
const int stat_mu = gsl_sf_bessel_Y_temme(mu, x, &Y_mu, &Y_mup1);
double Ynm1 = Y_mu.val;
double Yn = Y_mup1.val;
double Ynp1 = 0.0;
int n;
for(n=1; n<N; n++) {
Ynp1 = 2.0*(mu+n)/x * Yn - Ynm1;
Ynm1 = Yn;
Yn = Ynp1;
}
result->val = 2.0/(M_PI*x) / (Jnup1_Jnu*Yn - Ynp1);
result->err = GSL_DBL_EPSILON * (N + 2.0) * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_mu, stat_CF1);
}
else {
/* Recurse backward from nu to mu, determining the J ratio
* at mu. Use this together with a Steed method CF2 to
* determine the actual J_mu, and thus obtain the normalization.
*/
double Jmu;
double Jmup1_Jmu;
double sgn_Jmu;
double Jmuprime_Jmu;
double P, Q;
const int stat_CF2 = gsl_sf_bessel_JY_steed_CF2(mu, x, &P, &Q);
double gamma;
double Jnp1 = sgn_Jnu * GSL_SQRT_DBL_MIN * Jnup1_Jnu;
double Jn = sgn_Jnu * GSL_SQRT_DBL_MIN;
double Jnm1;
int n;
for(n=N; n>0; n--) {
Jnm1 = 2.0*(mu+n)/x * Jn - Jnp1;
Jnp1 = Jn;
Jn = Jnm1;
}
Jmup1_Jmu = Jnp1/Jn;
sgn_Jmu = GSL_SIGN(Jn);
Jmuprime_Jmu = mu/x - Jmup1_Jmu;
gamma = (P - Jmuprime_Jmu)/Q;
Jmu = sgn_Jmu * sqrt(2.0/(M_PI*x) / (Q + gamma*(P-Jmuprime_Jmu)));
result->val = Jmu * (sgn_Jnu * GSL_SQRT_DBL_MIN) / Jn;
result->err = 2.0 * GSL_DBL_EPSILON * (N + 2.0) * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_CF2, stat_CF1);
}
}
}
int
gsl_sf_bessel_Jnu_e(const double nu, const double x, gsl_sf_result * result)
{
/* CHECK_POINTER(result) */
if(x <= 0.0) {
DOMAIN_ERROR(result);
}
else if (nu < 0.0) {
int Jstatus = gsl_sf_bessel_Jnupos_e(-nu, x, result);
double Jval = result->val;
double Jerr = result->err;
int Ystatus = gsl_sf_bessel_Ynupos_e(-nu, x, result);
double Yval = result->val;
double Yerr = result->err;
/* double s = sin(M_PI*nu), c = cos(M_PI*nu); */
int sinstatus = gsl_sf_sin_pi_e(nu, result);
double s = result->val;
double serr = result->err;
int cosstatus = gsl_sf_cos_pi_e(nu, result);
double c = result->val;
double cerr = result->err;
result->val = s*Yval + c*Jval;
result->err = fabs(c*Yerr) + fabs(s*Jerr) + fabs(cerr*Yval) + fabs(serr*Jval);
return GSL_ERROR_SELECT_4(Jstatus, Ystatus, sinstatus, cosstatus);
}
else return gsl_sf_bessel_Jnupos_e(nu, x, result);
}
/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
#include "eval.h"
double gsl_sf_bessel_Jnu(const double nu, const double x)
{
EVAL_RESULT(gsl_sf_bessel_Jnu_e(nu, x, &result));
}
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