bessel_temme.c

开放gsl矩阵运算

/* 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 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., 675 Mass Ave, Cambridge, MA 02139, 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);
}