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bessel_I1.c

/* specfunc/bessel_I1.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_bessel.h>

#include "error.h"

#include "chebyshev.h"
#include "cheb_eval.c"

#define ROOT_EIGHT (2.0*M_SQRT2)


/*-*-*-*-*-*-*-*-*-*-*-* Private Section *-*-*-*-*-*-*-*-*-*-*-*/

/* based on SLATEC besi1(), besi1e() */

/* chebyshev expansions

 series for bi1        on the interval  0.          to  9.00000d+00
                                        with weighted error   2.40e-17
                                         log weighted error  16.62
                               significant figures required  16.23
                                    decimal places required  17.14

 series for ai1        on the interval  1.25000d-01 to  3.33333d-01
                                        with weighted error   6.98e-17
                                         log weighted error  16.16
                               significant figures required  14.53
                                    decimal places required  16.82

 series for ai12       on the interval  0.          to  1.25000d-01
                                       with weighted error   3.55e-17
                                        log weighted error  16.45
                              significant figures required  14.69
                                   decimal places required  17.12
*/

static double bi1_data[11] = {
  -0.001971713261099859,
   0.407348876675464810,
   0.034838994299959456,
   0.001545394556300123,
   0.000041888521098377,
   0.000000764902676483,
   0.000000010042493924,
   0.000000000099322077,
   0.000000000000766380,
   0.000000000000004741,
   0.000000000000000024
};
static cheb_series bi1_cs = {
  bi1_data,
  10,
  -1, 1,
  10
};

static double ai1_data[21] = {
  -0.02846744181881479,
  -0.01922953231443221,
  -0.00061151858579437,
  -0.00002069971253350,
   0.00000858561914581,
   0.00000104949824671,
  -0.00000029183389184,
  -0.00000001559378146,
   0.00000001318012367,
  -0.00000000144842341,
  -0.00000000029085122,
   0.00000000012663889,
  -0.00000000001664947,
  -0.00000000000166665,
   0.00000000000124260,
  -0.00000000000027315,
   0.00000000000002023,
   0.00000000000000730,
  -0.00000000000000333,
   0.00000000000000071,
  -0.00000000000000006
};
static cheb_series ai1_cs = {
  ai1_data,
  20,
  -1, 1,
  11
};

static double ai12_data[22] = {
   0.02857623501828014,
  -0.00976109749136147,
  -0.00011058893876263,
  -0.00000388256480887,
  -0.00000025122362377,
  -0.00000002631468847,
  -0.00000000383538039,
  -0.00000000055897433,
  -0.00000000001897495,
   0.00000000003252602,
   0.00000000001412580,
   0.00000000000203564,
  -0.00000000000071985,
  -0.00000000000040836,
  -0.00000000000002101,
   0.00000000000004273,
   0.00000000000001041,
  -0.00000000000000382,
  -0.00000000000000186,
   0.00000000000000033,
   0.00000000000000028,
  -0.00000000000000003
};
static cheb_series ai12_cs = {
  ai12_data,
  21,
  -1, 1,
  9
};


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

int gsl_sf_bessel_I1_scaled_e(const double x, gsl_sf_result * result)
{
  const double xmin    = 2.0 * GSL_DBL_MIN;
  const double x_small = ROOT_EIGHT * GSL_SQRT_DBL_EPSILON;
  const double y = fabs(x);

  /* CHECK_POINTER(result) */

  if(y == 0.0) {
    result->val = 0.0;
    result->err = 0.0;
    return GSL_SUCCESS;
  }
  else if(y < xmin) {
    UNDERFLOW_ERROR(result);
  }
  else if(y < x_small) {
    result->val = 0.5*x;
    result->err = 0.0;
    return GSL_SUCCESS;
  }
  else if(y <= 3.0) {
    const double ey = exp(-y);
    gsl_sf_result c;
    cheb_eval_e(&bi1_cs, y*y/4.5-1.0, &c);
    result->val  = x * ey * (0.875 + c.val);
    result->err  = ey * c.err + y * GSL_DBL_EPSILON * fabs(result->val);
    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return GSL_SUCCESS;
  }
  else if(y <= 8.0) {
    const double sy = sqrt(y);
    gsl_sf_result c;
    double b;
    double s;
    cheb_eval_e(&ai1_cs, (48.0/y-11.0)/5.0, &c);
    b = (0.375 + c.val) / sy;
    s = (x > 0.0 ? 1.0 : -1.0);
    result->val  = s * b;
    result->err  = c.err / sy;
    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return GSL_SUCCESS;
  }
  else {
    const double sy = sqrt(y);
    gsl_sf_result c;
    double b;
    double s;
    cheb_eval_e(&ai12_cs, 16.0/y-1.0, &c);
    b = (0.375 + c.val) / sy;
    s = (x > 0.0 ? 1.0 : -1.0);
    result->val  = s * b;
    result->err  = c.err / sy;
    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return GSL_SUCCESS;
  }
}


int gsl_sf_bessel_I1_e(const double x, gsl_sf_result * result)
{
  const double xmin    = 2.0 * GSL_DBL_MIN;
  const double x_small = ROOT_EIGHT * GSL_SQRT_DBL_EPSILON;
  const double y = fabs(x);

  /* CHECK_POINTER(result) */

  if(y == 0.0) {
    result->val = 0.0;
    result->err = 0.0;
    return GSL_SUCCESS;
  }
  else if(y < xmin) {
    UNDERFLOW_ERROR(result);
  }
  else if(y < x_small) {
    result->val = 0.5*x;
    result->err = 0.0;
    return GSL_SUCCESS;
  }
  else if(y <= 3.0) {
    gsl_sf_result c;
    cheb_eval_e(&bi1_cs, y*y/4.5-1.0, &c);
    result->val  = x * (0.875 + c.val);
    result->err  = y * c.err;
    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return GSL_SUCCESS;
  }
  else if(y < GSL_LOG_DBL_MAX) {
    const double ey = exp(y);
    gsl_sf_result I1_scaled;
    gsl_sf_bessel_I1_scaled_e(x, &I1_scaled);
    result->val  = ey * I1_scaled.val;
    result->err  = ey * I1_scaled.err + y * GSL_DBL_EPSILON * fabs(result->val);
    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return GSL_SUCCESS;
  }
  else {
    OVERFLOW_ERROR(result);
  }
}

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

#include "eval.h"

double gsl_sf_bessel_I1_scaled(const double x)
{
  EVAL_RESULT(gsl_sf_bessel_I1_scaled_e(x, &result));
}

double gsl_sf_bessel_I1(const double x)
{
  EVAL_RESULT(gsl_sf_bessel_I1_e(x, &result));
}

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