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/* specfunc/bessel_Yn.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_gamma.h>
#include <gsl/gsl_sf_psi.h>
#include <gsl/gsl_sf_bessel.h>
#include "error.h"
#include "bessel.h"
#include "bessel_amp_phase.h"
#include "bessel_olver.h"
/*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/
/* assumes n >= 1 */
static int bessel_Yn_small_x(const int n, const double x, gsl_sf_result * result)
{
int k;
double y = 0.25 * x * x;
double ln_x_2 = log(0.5*x);
gsl_sf_result ln_nm1_fact;
double k_term;
double term1, sum1, ln_pre1;
double term2, sum2, pre2;
gsl_sf_lnfact_e((unsigned int)(n-1), &ln_nm1_fact);
ln_pre1 = -n*ln_x_2 + ln_nm1_fact.val;
if(ln_pre1 > GSL_LOG_DBL_MAX - 3.0) GSL_ERROR ("error", GSL_EOVRFLW);
sum1 = 1.0;
k_term = 1.0;
for(k=1; k<=n-1; k++) {
k_term *= y/(k * (n-k));
sum1 += k_term;
}
term1 = -exp(ln_pre1) * sum1 / M_PI;
pre2 = -exp(n*ln_x_2) / M_PI;
if(fabs(pre2) > 0.0) {
const int KMAX = 20;
gsl_sf_result psi_n;
gsl_sf_result npk_fact;
double yk = 1.0;
double k_fact = 1.0;
double psi_kp1 = -M_EULER;
double psi_npkp1;
gsl_sf_psi_int_e(n, &psi_n);
gsl_sf_fact_e((unsigned int)n, &npk_fact);
psi_npkp1 = psi_n.val + 1.0/n;
sum2 = (psi_kp1 + psi_npkp1 - 2.0*ln_x_2)/npk_fact.val;
for(k=1; k<KMAX; k++) {
psi_kp1 += 1./k;
psi_npkp1 += 1./(n+k);
k_fact *= k;
npk_fact.val *= n+k;
yk *= -y;
k_term = yk*(psi_kp1 + psi_npkp1 - 2.0*ln_x_2)/(k_fact*npk_fact.val);
sum2 += k_term;
}
term2 = pre2 * sum2;
}
else {
term2 = 0.0;
}
result->val = term1 + term2;
result->err = GSL_DBL_EPSILON * (fabs(ln_pre1)*fabs(term1) + fabs(term2));
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_SUCCESS;
}
/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/
int
gsl_sf_bessel_Yn_e(int n, const double x, gsl_sf_result * result)
{
int sign = 1;
if(n < 0) {
/* reduce to case n >= 0 */
n = -n;
if(GSL_IS_ODD(n)) sign = -1;
}
/* CHECK_POINTER(result) */
if(n == 0) {
int status = gsl_sf_bessel_Y0_e(x, result);
result->val *= sign;
return status;
}
else if(n == 1) {
int status = gsl_sf_bessel_Y1_e(x, result);
result->val *= sign;
return status;
}
else {
if(x <= 0.0) {
DOMAIN_ERROR(result);
}
if(x < 5.0) {
int status = bessel_Yn_small_x(n, x, result);
result->val *= sign;
return status;
}
else if(GSL_ROOT3_DBL_EPSILON * x > (n*n + 1.0)) {
int status = gsl_sf_bessel_Ynu_asympx_e((double)n, x, result);
result->val *= sign;
return status;
}
else if(n > 50) {
int status = gsl_sf_bessel_Ynu_asymp_Olver_e((double)n, x, result);
result->val *= sign;
return status;
}
else {
double two_over_x = 2.0/x;
gsl_sf_result r_by;
gsl_sf_result r_bym;
int stat_1 = gsl_sf_bessel_Y1_e(x, &r_by);
int stat_0 = gsl_sf_bessel_Y0_e(x, &r_bym);
double bym = r_bym.val;
double by = r_by.val;
double byp;
int j;
for(j=1; j<n; j++) {
byp = j*two_over_x*by - bym;
bym = by;
by = byp;
}
result->val = sign * by;
result->err = fabs(result->val) * (fabs(r_by.err/r_by.val) + fabs(r_bym.err/r_bym.val));
result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
return GSL_ERROR_SELECT_2(stat_1, stat_0);
}
}
}
int
gsl_sf_bessel_Yn_array(const int nmin, const int nmax, const double x, double * result_array)
{
/* CHECK_POINTER(result_array) */
if(nmin < 0 || nmax < nmin || x <= 0.0) {
int j;
for(j=0; j<=nmax-nmin; j++) result_array[j] = 0.0;
GSL_ERROR ("error", GSL_EDOM);
}
else {
gsl_sf_result r_Ynm1;
gsl_sf_result r_Yn;
int stat_nm1 = gsl_sf_bessel_Yn_e(nmin, x, &r_Ynm1);
int stat_n = gsl_sf_bessel_Yn_e(nmin+1, x, &r_Yn);
double Ynp1;
double Yn = r_Yn.val;
double Ynm1 = r_Ynm1.val;
int n;
int stat = GSL_ERROR_SELECT_2(stat_nm1, stat_n);
if(stat == GSL_SUCCESS) {
for(n=nmin+1; n<=nmax+1; n++) {
result_array[n-nmin-1] = Ynm1;
Ynp1 = -Ynm1 + 2.0*n/x * Yn;
Ynm1 = Yn;
Yn = Ynp1;
}
}
else {
for(n=nmin; n<=nmax; n++) {
result_array[n-nmin] = 0.0;
}
}
return stat;
}
}
/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/
#include "eval.h"
double gsl_sf_bessel_Yn(const int n, const double x)
{
EVAL_RESULT(gsl_sf_bessel_Yn_e(n, x, &result));
}
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