text: /* 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., 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);
  }
  