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bessel_i.c

/* specfunc/bessel_i.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_pow_int.h>
#include <gsl/gsl_sf_bessel.h>

#include "error.h"

#include "bessel.h"


/* i_{l+1}/i_l
 */
static
int
bessel_il_CF1(const int l, const double x, const double threshold, double * ratio)
{
  const int kmax = 2000;
  double tk   = 1.0;
  double sum  = 1.0;
  double rhok = 0.0;
  int k;

  for(k=1; k<=kmax; k++) {
    double ak = (x/(2.0*l+1.0+2.0*k)) * (x/(2.0*l+3.0+2.0*k));
    rhok = -ak*(1.0 + rhok)/(1.0 + ak*(1.0 + rhok));
    tk  *= rhok;
    sum += tk;
    if(fabs(tk/sum) < threshold) break;
  }

  *ratio = x/(2.0*l+3.0) * sum;

  if(k == kmax)
    GSL_ERROR ("error", GSL_EMAXITER);
  else
    return GSL_SUCCESS;
}


/*-*-*-*-*-*-*-*-*-*-*-* Functions with Error Codes *-*-*-*-*-*-*-*-*-*-*-*/

int gsl_sf_bessel_i0_scaled_e(const double x, gsl_sf_result * result)
{
  double ax = fabs(x);

  /* CHECK_POINTER(result) */

  if(x == 0.0) {
    result->val = 1.0;
    result->err = 0.0;
    return GSL_SUCCESS;    
  }
  else if(ax < 0.2) {
    const double eax = exp(-ax);
    const double y = ax*ax;
    const double c1 = 1.0/6.0;
    const double c2 = 1.0/120.0;
    const double c3 = 1.0/5040.0;
    const double c4 = 1.0/362880.0;
    const double c5 = 1.0/39916800.0;
    const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*c5))));
    result->val = eax * sum;
    result->err = 2.0 * GSL_DBL_EPSILON * result->val;
  }
  else if(ax < -0.5*GSL_LOG_DBL_EPSILON) {
    result->val = (1.0 - exp(-2.0*ax))/(2.0*ax);
    result->err = 2.0 * GSL_DBL_EPSILON * result->val;
  }
  else {
    result->val = 1.0/(2.0*ax);
    result->err = 2.0 * GSL_DBL_EPSILON * result->val;
  }
  return GSL_SUCCESS;
}


int gsl_sf_bessel_i1_scaled_e(const double x, gsl_sf_result * result)
{
  double ax = fabs(x);

  /* CHECK_POINTER(result) */

  if(x == 0.0) {
    result->val = 0.0;
    result->err = 0.0;
    return GSL_SUCCESS;
  }
  else if(ax < 3.0*GSL_DBL_MIN) {
    UNDERFLOW_ERROR(result);
  }
  else if(ax < 0.25) {
    const double eax = exp(-ax);
    const double y  = x*x;
    const double c1 = 1.0/10.0;
    const double c2 = 1.0/280.0;
    const double c3 = 1.0/15120.0;
    const double c4 = 1.0/1330560.0;
    const double c5 = 1.0/172972800.0;
    const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*c5))));
    result->val = eax * x/3.0 * sum;
    result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return GSL_SUCCESS;
  }
  else {
    double ex = exp(-2.0*ax);
    result->val = 0.5 * (ax*(1.0+ex) - (1.0-ex)) / (ax*ax);
    result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    if(x < 0.0) result->val = -result->val;
    return GSL_SUCCESS;
  }
}


int gsl_sf_bessel_i2_scaled_e(const double x, gsl_sf_result * result)
{
  double ax = fabs(x);

  /* CHECK_POINTER(result) */

  if(x == 0.0) {
    result->val = 0.0;
    result->err = 0.0;
    return GSL_SUCCESS;    
  }
  else if(ax < 4.0*GSL_SQRT_DBL_MIN) {
    UNDERFLOW_ERROR(result);
  }
  else if(ax < 0.25) {
    const double y = x*x;
    const double c1 = 1.0/14.0;
    const double c2 = 1.0/504.0;
    const double c3 = 1.0/33264.0;
    const double c4 = 1.0/3459456.0;
    const double c5 = 1.0/518918400.0;
    const double sum = 1.0 + y*(c1 + y*(c2 + y*(c3 + y*(c4 + y*c5))));
    const double pre = exp(-ax) * x*x/15.0;
    result->val = pre * sum;
    result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return GSL_SUCCESS;
  }
  else {
    double ex = exp(-2.0*ax);
    double x2 = x*x;
    result->val = 0.5 * ((3.0+x2)*(1.0-ex) - 3.0*ax*(1.0+ex))/(ax*ax*ax);
    result->err = 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return GSL_SUCCESS;
  }
}  


int gsl_sf_bessel_il_scaled_e(const int l, double x, gsl_sf_result * result)
{
  double sgn = 1.0;
  double ax  = fabs(x);

  if(x < 0.0) {
    /* i_l(-x) = (-1)^l i_l(x) */
    sgn = ( GSL_IS_ODD(l) ? -1.0 : 1.0 );
    x = -x;
  }

  if(l < 0) {
    DOMAIN_ERROR(result);
  }
  else if(x == 0.0) {
    result->val = ( l == 0 ? 1.0 : 0.0 );
    result->err = 0.0;
    return GSL_SUCCESS;
  }
  else if(l == 0) {
    gsl_sf_result il;
    int stat_il = gsl_sf_bessel_i0_scaled_e(x, &il);
    result->val = sgn * il.val;
    result->err = il.err;
    return stat_il;
  }
  else if(l == 1) {
    gsl_sf_result il;
    int stat_il = gsl_sf_bessel_i1_scaled_e(x, &il);
    result->val = sgn * il.val;
    result->err = il.err;
    return stat_il;
  }
  else if(l == 2) {
    gsl_sf_result il;
    int stat_il = gsl_sf_bessel_i2_scaled_e(x, &il);
    result->val = sgn * il.val;
    result->err = il.err;
    return stat_il;
  }
  else if(x*x < 10.0*(l+1.5)/M_E) {
    gsl_sf_result b;
    int stat = gsl_sf_bessel_IJ_taylor_e(l+0.5, x, 1, 50, GSL_DBL_EPSILON, &b);
    double pre   = exp(-ax) * sqrt((0.5*M_PI)/x);
    result->val  = sgn * pre * b.val;
    result->err  = pre * b.err;
    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return stat;
  }
  else if(l < 150) {
    gsl_sf_result i0_scaled;
    int stat_i0  = gsl_sf_bessel_i0_scaled_e(ax, &i0_scaled);
    double rat;
    int stat_CF1 = bessel_il_CF1(l, ax, GSL_DBL_EPSILON, &rat);
    double iellp1 = rat * GSL_SQRT_DBL_MIN;
    double iell   = GSL_SQRT_DBL_MIN;
    double iellm1;
    int ell;
    for(ell = l; ell >= 1; ell--) {
      iellm1 = iellp1 + (2*ell + 1)/x * iell;
      iellp1 = iell;
      iell   = iellm1;
    }
    result->val  = sgn * i0_scaled.val * (GSL_SQRT_DBL_MIN / iell);
    result->err  = i0_scaled.err * (GSL_SQRT_DBL_MIN / iell);
    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return GSL_ERROR_SELECT_2(stat_i0, stat_CF1);
  }
  else if(GSL_MIN(0.29/(l*l+1.0), 0.5/(l*l+1.0+x*x)) < 0.5*GSL_ROOT3_DBL_EPSILON) {
    int status = gsl_sf_bessel_Inu_scaled_asymp_unif_e(l + 0.5, x, result);
    double pre = sqrt((0.5*M_PI)/x);
    result->val *= sgn * pre;
    result->err *= pre;
    return status;
  }
  else {
    /* recurse down from safe values */
    double rt_term = sqrt((0.5*M_PI)/x);
    const int LMAX = 2 + (int) (1.2 / GSL_ROOT6_DBL_EPSILON);
    gsl_sf_result r_iellp1;
    gsl_sf_result r_iell;
    int stat_a1 = gsl_sf_bessel_Inu_scaled_asymp_unif_e(LMAX + 1 + 0.5, x, &r_iellp1);
    int stat_a2 = gsl_sf_bessel_Inu_scaled_asymp_unif_e(LMAX     + 0.5, x, &r_iell);
    double iellp1 = r_iellp1.val;
    double iell   = r_iell.val;
    double iellm1 = 0.0;
    int ell;
    iellp1 *= rt_term;
    iell   *= rt_term;
    for(ell = LMAX; ell >= l+1; ell--) {
      iellm1 = iellp1 + (2*ell + 1)/x * iell;
      iellp1 = iell;
      iell   = iellm1;
    }
    result->val  = sgn * iellm1;
    result->err  = fabs(result->val)*(GSL_DBL_EPSILON + fabs(r_iellp1.err/r_iellp1.val) + fabs(r_iell.err/r_iell.val));
    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);

    return GSL_ERROR_SELECT_2(stat_a1, stat_a2);
  }
}


int gsl_sf_bessel_il_scaled_array(const int lmax, const double x, double * result_array)
{
  if(x == 0.0) {
    int ell;
    result_array[0] = 1.0;
    for (ell = lmax; ell >= 1; ell--) {
      result_array[ell] = 0.0;
    };
    return GSL_SUCCESS;
  } else {
    int ell;
    gsl_sf_result r_iellp1;
    gsl_sf_result r_iell;
    int stat_0 = gsl_sf_bessel_il_scaled_e(lmax+1, x, &r_iellp1);
    int stat_1 = gsl_sf_bessel_il_scaled_e(lmax,   x, &r_iell);
    double iellp1 = r_iellp1.val;
    double iell   = r_iell.val;
    double iellm1;
    result_array[lmax] = iell;
    for(ell = lmax; ell >= 1; ell--) {
      iellm1 = iellp1 + (2*ell + 1)/x * iell;
      iellp1 = iell;
      iell   = iellm1;
      result_array[ell-1] = iellm1;
    }
    return GSL_ERROR_SELECT_2(stat_0, stat_1);
  }
}


/*-*-*-*-*-*-*-*-*-* Functions w/ Natural Prototypes *-*-*-*-*-*-*-*-*-*-*/

#include "eval.h"

double gsl_sf_bessel_i0_scaled(const double x)
{
  EVAL_RESULT(gsl_sf_bessel_i0_scaled_e(x, &result));
}

double gsl_sf_bessel_i1_scaled(const double x)
{
  EVAL_RESULT(gsl_sf_bessel_i1_scaled_e(x, &result));
}

double gsl_sf_bessel_i2_scaled(const double x)
{
  EVAL_RESULT(gsl_sf_bessel_i2_scaled_e(x, &result));
}

double gsl_sf_bessel_il_scaled(const int l, const double x)
{
  EVAL_RESULT(gsl_sf_bessel_il_scaled_e(l, x, &result));
}


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