/* specfunc/bessel_temme.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 */
/* Calculate series for Y_nu and K_nu for small x and nu.
* This is applicable for x < 2 and |nu|<=1/2.
* These functions assume x > 0.
*/
#include <config.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_mode.h>
#include "bessel_temme.h"
#include "chebyshev.h"
#include "cheb_eval.c"
/* nu = (x+1)/4, -1<x<1, 1/(2nu)(1/Gamma[1-nu]-1/Gamma[1+nu]) */
static double g1_dat[14] = {
-1.14516408366268311786898152867,
0.00636085311347084238122955495,
0.00186245193007206848934643657,
0.000152833085873453507081227824,
0.000017017464011802038795324732,
-6.4597502923347254354668326451e-07,
-5.1819848432519380894104312968e-08,
4.5189092894858183051123180797e-10,
3.2433227371020873043666259180e-11,
6.8309434024947522875432400828e-13,
2.8353502755172101513119628130e-14,
-7.9883905769323592875638087541e-16,
-3.3726677300771949833341213457e-17,
-3.6586334809210520744054437104e-20
};
static cheb_series g1_cs = {
g1_dat,
13,
-1, 1,
7
};
/* nu = (x+1)/4, -1<x<1, 1/2 (1/Gamma[1-nu]+1/Gamma[1+nu]) */
static double g2_dat[15] =
{
1.882645524949671835019616975350,
-0.077490658396167518329547945212,
-0.018256714847324929419579340950,
0.0006338030209074895795923971731,
0.0000762290543508729021194461175,
-9.5501647561720443519853993526e-07,
-8.8927268107886351912431512955e-08,
-1.9521334772319613740511880132e-09,
-9.4003052735885162111769579771e-11,
4.6875133849532393179290879101e-12,
2.2658535746925759582447545145e-13,
-1.1725509698488015111878735251e-15,
-7.0441338200245222530843155877e-17,
-2.4377878310107693650659740228e-18,
-7.5225243218253901727164675011e-20
};
static cheb_series g2_cs = {
g2_dat,
14,
-1, 1,
8
};
static
int
gsl_sf_temme_gamma(const double nu, double * g_1pnu, double * g_1mnu, double * g1, double * g2)
{
const double anu = fabs(nu); /* functions are even */
const double x = 4.0*anu - 1.0;
gsl_sf_result r_g1;
gsl_sf_result r_g2;
cheb_eval_e(&g1_cs, x, &r_g1);
cheb_eval_e(&g2_cs, x, &r_g2);
*g1 = r_g1.val;
*g2 = r_g2.val;
*g_1mnu = 1.0/(r_g2.val + nu * r_g1.val);
*g_1pnu = 1.0/(r_g2.val - nu * r_g1.val);
return GSL_SUCCESS;
}
int
gsl_sf_bessel_Y_temme(const double nu, const double x,
gsl_sf_result * Ynu,
gsl_sf_result * Ynup1)
{
const int max_iter = 15000;
const double half_x = 0.5 * x;
const double ln_half_x = log(half_x);
const double half_x_nu = exp(nu*ln_half_x);
const double pi_nu = M_PI * nu;
const double alpha = pi_nu / 2.0;
const double sigma = -nu * ln_half_x;
const double sinrat = (fabs(pi_nu) < GSL_DBL_EPSILON ? 1.0 : pi_nu/sin(pi_nu));
const double sinhrat = (fabs(sigma) < GSL_DBL_EPSILON ? 1.0 : sinh(sigma)/sigma);
const double sinhalf = (fabs(alpha) < GSL_DBL_EPSILON ? 1.0 : sin(alpha)/alpha);
const double sin_sqr = nu*M_PI*M_PI*0.5 * sinhalf*sinhalf;
double sum0, sum1;
double fk, pk, qk, hk, ck;
int k = 0;
int stat_iter;
double g_1pnu, g_1mnu, g1, g2;
int stat_g = gsl_sf_temme_gamma(nu, &g_1pnu, &g_1mnu, &g1, &g2);
fk = 2.0/M_PI * sinrat * (cosh(sigma)*g1 - sinhrat*ln_half_x*g2);
pk = 1.0/M_PI /half_x_nu * g_1pnu;
qk = 1.0/M_PI *half_x_nu * g_1mnu;
hk = pk;
ck = 1.0;
sum0 = fk + sin_sqr * qk;
sum1 = pk;
while(k < max_iter) {
double del0;
double del1;
double gk;
k++;
fk = (k*fk + pk + qk)/(k*k-nu*nu);
ck *= -half_x*half_x/k;
pk /= (k - nu);
qk /= (k + nu);
gk = fk + sin_sqr * qk;
hk = -k*gk + pk;
del0 = ck * gk;
del1 = ck * hk;
sum0 += del0;
sum1 += del1;
if(fabs(del0) < 0.5*(1.0 + fabs(sum0))*GSL_DBL_EPSILON) break;
}
Ynu->val = -sum0;
Ynu->err = (2.0 + 0.5*k) * GSL_DBL_EPSILON * fabs(Ynu->val);
Ynup1->val = -sum1 * 2.0/x;
Ynup1->err = (2.0 + 0.5*k) * GSL_DBL_EPSILON * fabs(Ynup1->val);
stat_iter = ( k >= max_iter ? GSL_EMAXITER : GSL_SUCCESS );
return GSL_ERROR_SELECT_2(stat_iter, stat_g);
}
int
gsl_sf_bessel_K_scaled_temme(const double nu, const double x,
double * K_nu, double * K_nup1, double * Kp_nu)
{
const int max_iter = 15000;
const double half_x = 0.5 * x;
const double ln_half_x = log(half_x);
const double half_x_nu = exp(nu*ln_half_x);
const double pi_nu = M_PI * nu;
const double sigma = -nu * ln_half_x;
const double sinrat = (fabs(pi_nu) < GSL_DBL_EPSILON ? 1.0 : pi_nu/sin(pi_nu));
const double sinhrat = (fabs(sigma) < GSL_DBL_EPSILON ? 1.0 : sinh(sigma)/sigma);
const double ex = exp(x);
double sum0, sum1;
double fk, pk, qk, hk, ck;
int k = 0;
int stat_iter;
double g_1pnu, g_1mnu, g1, g2;
int stat_g = gsl_sf_temme_gamma(nu, &g_1pnu, &g_1mnu, &g1, &g2);
fk = sinrat * (cosh(sigma)*g1 - sinhrat*ln_half_x*g2);
pk = 0.5/half_x_nu * g_1pnu;
qk = 0.5*half_x_nu * g_1mnu;
hk = pk;
ck = 1.0;
sum0 = fk;
sum1 = hk;
while(k < max_iter) {
double del0;
double del1;
k++;
fk = (k*fk + pk + qk)/(k*k-nu*nu);
ck *= half_x*half_x/k;
pk /= (k - nu);
qk /= (k + nu);
hk = -k*fk + pk;
del0 = ck * fk;
del1 = ck * hk;
sum0 += del0;
sum1 += del1;
if(fabs(del0) < 0.5*fabs(sum0)*GSL_DBL_EPSILON) break;
}
*K_nu = sum0 * ex;
*K_nup1 = sum1 * 2.0/x * ex;
*Kp_nu = - *K_nup1 + nu/x * *K_nu;
stat_iter = ( k == max_iter ? GSL_EMAXITER : GSL_SUCCESS );
return GSL_ERROR_SELECT_2(stat_iter, stat_g);
}