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expint.c

/* specfunc/expint.c
 * 
 * Copyright (C) 2007 Brian Gough
 * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002 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 */

#include <config.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_sf_expint.h>
#include <gsl/gsl_sf_gamma.h>

#include "error.h"
#include "check.h"

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

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

/*
 Chebyshev expansions: based on SLATEC e1.f, W. Fullerton
 
 Series for AE11       on the interval -1.00000D-01 to  0.
                                        with weighted error   1.76E-17
                                         log weighted error  16.75
                               significant figures required  15.70
                                    decimal places required  17.55


 Series for AE12       on the interval -2.50000D-01 to -1.00000D-01
                                        with weighted error   5.83E-17
                                         log weighted error  16.23
                               significant figures required  15.76
                                    decimal places required  16.93


 Series for E11        on the interval -4.00000D+00 to -1.00000D+00
                                        with weighted error   1.08E-18
                                         log weighted error  17.97
                               significant figures required  19.02
                                    decimal places required  18.61


 Series for E12        on the interval -1.00000D+00 to  1.00000D+00
                                        with weighted error   3.15E-18
                                         log weighted error  17.50
                        approx significant figures required  15.8
                                    decimal places required  18.10


 Series for AE13       on the interval  2.50000D-01 to  1.00000D+00
                                        with weighted error   2.34E-17
                                         log weighted error  16.63
                               significant figures required  16.14
                                    decimal places required  17.33


 Series for AE14       on the interval  0.          to  2.50000D-01
                                        with weighted error   5.41E-17
                                         log weighted error  16.27
                               significant figures required  15.38
                                    decimal places required  16.97
*/

static double AE11_data[39] = {
   0.121503239716065790,
  -0.065088778513550150,
   0.004897651357459670,
  -0.000649237843027216,
   0.000093840434587471,
   0.000000420236380882,
  -0.000008113374735904,
   0.000002804247688663,
   0.000000056487164441,
  -0.000000344809174450,
   0.000000058209273578,
   0.000000038711426349,
  -0.000000012453235014,
  -0.000000005118504888,
   0.000000002148771527,
   0.000000000868459898,
  -0.000000000343650105,
  -0.000000000179796603,
   0.000000000047442060,
   0.000000000040423282,
  -0.000000000003543928,
  -0.000000000008853444,
  -0.000000000000960151,
   0.000000000001692921,
   0.000000000000607990,
  -0.000000000000224338,
  -0.000000000000200327,
  -0.000000000000006246,
   0.000000000000045571,
   0.000000000000016383,
  -0.000000000000005561,
  -0.000000000000006074,
  -0.000000000000000862,
   0.000000000000001223,
   0.000000000000000716,
  -0.000000000000000024,
  -0.000000000000000201,
  -0.000000000000000082,
   0.000000000000000017
};
static cheb_series AE11_cs = {
  AE11_data,
  38,
  -1, 1,
  20
};

static double AE12_data[25] = {
   0.582417495134726740,
  -0.158348850905782750,
  -0.006764275590323141,
   0.005125843950185725,
   0.000435232492169391,
  -0.000143613366305483,
  -0.000041801320556301,
  -0.000002713395758640,
   0.000001151381913647,
   0.000000420650022012,
   0.000000066581901391,
   0.000000000662143777,
  -0.000000002844104870,
  -0.000000000940724197,
  -0.000000000177476602,
  -0.000000000015830222,
   0.000000000002905732,
   0.000000000001769356,
   0.000000000000492735,
   0.000000000000093709,
   0.000000000000010707,
  -0.000000000000000537,
  -0.000000000000000716,
  -0.000000000000000244,
  -0.000000000000000058
};
static cheb_series AE12_cs = {
  AE12_data,
  24,
  -1, 1,
  15
};

static double E11_data[19] = {
  -16.11346165557149402600,
    7.79407277874268027690,
   -1.95540581886314195070,
    0.37337293866277945612,
   -0.05692503191092901938,
    0.00721107776966009185,
   -0.00078104901449841593,
    0.00007388093356262168,
   -0.00000620286187580820,
    0.00000046816002303176,
   -0.00000003209288853329,
    0.00000000201519974874,
   -0.00000000011673686816,
    0.00000000000627627066,
   -0.00000000000031481541,
    0.00000000000001479904,
   -0.00000000000000065457,
    0.00000000000000002733,
   -0.00000000000000000108
};
static cheb_series E11_cs = {
  E11_data,
  18,
  -1, 1,
  13
};

static double E12_data[16] = {
  -0.03739021479220279500,
   0.04272398606220957700,
  -0.13031820798497005440,
   0.01441912402469889073,
  -0.00134617078051068022,
   0.00010731029253063780,
  -0.00000742999951611943,
   0.00000045377325690753,
  -0.00000002476417211390,
   0.00000000122076581374,
  -0.00000000005485141480,
   0.00000000000226362142,
  -0.00000000000008635897,
   0.00000000000000306291,
  -0.00000000000000010148,
   0.00000000000000000315
};
static cheb_series E12_cs = {
  E12_data,
  15,
  -1, 1,
  10
};

static double AE13_data[25] = {
  -0.605773246640603460,
  -0.112535243483660900,
   0.013432266247902779,
  -0.001926845187381145,
   0.000309118337720603,
  -0.000053564132129618,
   0.000009827812880247,
  -0.000001885368984916,
   0.000000374943193568,
  -0.000000076823455870,
   0.000000016143270567,
  -0.000000003466802211,
   0.000000000758754209,
  -0.000000000168864333,
   0.000000000038145706,
  -0.000000000008733026,
   0.000000000002023672,
  -0.000000000000474132,
   0.000000000000112211,
  -0.000000000000026804,
   0.000000000000006457,
  -0.000000000000001568,
   0.000000000000000383,
  -0.000000000000000094,
   0.000000000000000023
};
static cheb_series AE13_cs = {
  AE13_data,
  24,
  -1, 1,
  15
};

static double AE14_data[26] = {
  -0.18929180007530170,
  -0.08648117855259871,
   0.00722410154374659,
  -0.00080975594575573,
   0.00010999134432661,
  -0.00001717332998937,
   0.00000298562751447,
  -0.00000056596491457,
   0.00000011526808397,
  -0.00000002495030440,
   0.00000000569232420,
  -0.00000000135995766,
   0.00000000033846628,
  -0.00000000008737853,
   0.00000000002331588,
  -0.00000000000641148,
   0.00000000000181224,
  -0.00000000000052538,
   0.00000000000015592,
  -0.00000000000004729,
   0.00000000000001463,
  -0.00000000000000461,
   0.00000000000000148,
  -0.00000000000000048,
   0.00000000000000016,
  -0.00000000000000005
};
static cheb_series AE14_cs = {
  AE14_data,
  25,
  -1, 1,
  13
};



/* implementation for E1, allowing for scaling by exp(x) */
static
int expint_E1_impl(const double x, gsl_sf_result * result, const int scale)
{
  const double xmaxt = -GSL_LOG_DBL_MIN;      /* XMAXT = -LOG (R1MACH(1)) */
  const double xmax  = xmaxt - log(xmaxt);    /* XMAX = XMAXT - LOG(XMAXT) */

  /* CHECK_POINTER(result) */

  if(x < -xmax && !scale) {
      OVERFLOW_ERROR(result);
  }
  else if(x <= -10.0) {
    const double s = 1.0/x * ( scale ? 1.0 : exp(-x) );
    gsl_sf_result result_c;
    cheb_eval_e(&AE11_cs, 20.0/x+1.0, &result_c);
    result->val  = s * (1.0 + result_c.val);
    result->err  = s * result_c.err;
    result->err += 2.0 * GSL_DBL_EPSILON * (fabs(x) + 1.0) * fabs(result->val);
    return GSL_SUCCESS;
  }
  else if(x <= -4.0) {
    const double s = 1.0/x * ( scale ? 1.0 : exp(-x) );
    gsl_sf_result result_c;
    cheb_eval_e(&AE12_cs, (40.0/x+7.0)/3.0, &result_c);
    result->val  = s * (1.0 + result_c.val);
    result->err  = s * result_c.err;
    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return GSL_SUCCESS;
  }
  else if(x <= -1.0) {
    const double ln_term = -log(fabs(x));
    const double scale_factor = ( scale ? exp(x) : 1.0 );
    gsl_sf_result result_c;
    cheb_eval_e(&E11_cs, (2.0*x+5.0)/3.0, &result_c);
    result->val  = scale_factor * (ln_term + result_c.val);
    result->err  = scale_factor * (result_c.err + GSL_DBL_EPSILON * fabs(ln_term));
    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return GSL_SUCCESS;
  }
  else if(x == 0.0) {
    DOMAIN_ERROR(result);
  }
  else if(x <= 1.0) {
    const double ln_term = -log(fabs(x));
    const double scale_factor = ( scale ? exp(x) : 1.0 );
    gsl_sf_result result_c;
    cheb_eval_e(&E12_cs, x, &result_c);
    result->val  = scale_factor * (ln_term - 0.6875 + x + result_c.val);
    result->err  = scale_factor * (result_c.err + GSL_DBL_EPSILON * fabs(ln_term));
    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return GSL_SUCCESS;
  }
  else if(x <= 4.0) {
    const double s = 1.0/x * ( scale ? 1.0 : exp(-x) );
    gsl_sf_result result_c;
    cheb_eval_e(&AE13_cs, (8.0/x-5.0)/3.0, &result_c);
    result->val  = s * (1.0 + result_c.val);
    result->err  = s * result_c.err;
    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return GSL_SUCCESS;
  }
  else if(x <= xmax || scale) {
    const double s = 1.0/x * ( scale ? 1.0 : exp(-x) );
    gsl_sf_result result_c;
    cheb_eval_e(&AE14_cs, 8.0/x-1.0, &result_c);
    result->val  = s * (1.0 +  result_c.val);
    result->err  = s * (GSL_DBL_EPSILON + result_c.err);
    result->err += 2.0 * (x + 1.0) * GSL_DBL_EPSILON * fabs(result->val);
    if(result->val == 0.0)
      UNDERFLOW_ERROR(result);
    else
      return GSL_SUCCESS;
  }
  else {
    UNDERFLOW_ERROR(result);
  }
}


static
int expint_E2_impl(const double x, gsl_sf_result * result, const int scale)
{
  const double xmaxt = -GSL_LOG_DBL_MIN;
  const double xmax  = xmaxt - log(xmaxt);

  /* CHECK_POINTER(result) */

  if(x < -xmax && !scale) {
    OVERFLOW_ERROR(result);
  }
  else if (x == 0.0) {
    result->val = (scale ? 1.0 : 1.0);
    result->err = 0.0;
    return GSL_SUCCESS;
  } else if(x < 100.0) {
    const double ex = ( scale ? 1.0 : exp(-x) );
    gsl_sf_result result_E1;
    int stat_E1 = expint_E1_impl(x, &result_E1, scale);
    result->val  = ex - x*result_E1.val;
    result->err  = GSL_DBL_EPSILON*ex + fabs(x) * result_E1.err;
    result->err += 2.0 * GSL_DBL_EPSILON * fabs(result->val);
    return stat_E1;
  }
  else if(x < xmax || scale) {
    const double s = ( scale ? 1.0 : exp(-x) );
    const double c1  = -2.0;
    const double c2  =  6.0;
    const double c3  = -24.0;
    const double c4  =  120.0;
    const double c5  = -720.0;
    const double c6  =  5040.0;
    const double c7  = -40320.0;
    const double c8  =  362880.0;
    const double c9  = -3628800.0;
    const double c10 =  39916800.0;
    const double c11 = -479001600.0;
    const double c12 =  6227020800.0;
    const double c13 = -87178291200.0;
    const double y = 1.0/x;
    const double sum6 = c6+y*(c7+y*(c8+y*(c9+y*(c10+y*(c11+y*(c12+y*c13))))));
    const double sum  = y*(c1+y*(c2+y*(c3+y*(c4+y*(c5+y*sum6)))));
    result->val = s * (1.0 + sum)/x;
    result->err = 2.0 * (x + 1.0) * GSL_DBL_EPSILON * result->val;
    if(result->val == 0.0)
      UNDERFLOW_ERROR(result);
    else
      return GSL_SUCCESS;
  }
  else {
    UNDERFLOW_ERROR(result);
  }
}

static
int expint_En_impl(const int n, const double x, gsl_sf_result * result, const int scale)
{
  if (n < 0) {
    DOMAIN_ERROR(result);
  } else if (n == 0) {
    if (x == 0) {
      DOMAIN_ERROR(result);
    } else {
      result->val = (scale ? 1.0 : exp(-x)) / x;
      result->err = 2 * GSL_DBL_EPSILON * fabs(result->val);
      CHECK_UNDERFLOW(result);
      return GSL_SUCCESS;
    }
  } else if (n == 1) {
    return expint_E1_impl(x, result, scale);
  } else if (n == 2) {
    return expint_E2_impl(x, result, scale);
  } else { 
    if(x < 0) {
      DOMAIN_ERROR(result);
    }
    if (x == 0) {
      result->val = (scale ? exp(x) : 1 ) * (1/(n-1.0));
      result->err = 2 * GSL_DBL_EPSILON * fabs(result->val);
      CHECK_UNDERFLOW(result);
      return GSL_SUCCESS;
    } else {
      gsl_sf_result result_g;
      double prefactor = pow(x, n-1);
      int status = gsl_sf_gamma_inc_e (1-n, x, &result_g);
      double scale_factor = ( scale ? exp(x) : 1.0 );
      result->val = scale_factor * prefactor * result_g.val;
      result->err = 2 * GSL_DBL_EPSILON * fabs(result->val);
      result->err += 2 * fabs(scale_factor * prefactor * result_g.err);
      if (status == GSL_SUCCESS) CHECK_UNDERFLOW(result);
      return status;
    }
  }
}

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


int gsl_sf_expint_E1_e(const double x, gsl_sf_result * result)
{
  return expint_E1_impl(x, result, 0);
}


int gsl_sf_expint_E1_scaled_e(const double x, gsl_sf_result * result)
{
  return expint_E1_impl(x, result, 1);
}


int gsl_sf_expint_E2_e(const double x, gsl_sf_result * result)
{
  return expint_E2_impl(x, result, 0);
}


int gsl_sf_expint_E2_scaled_e(const double x, gsl_sf_result * result)
{
  return expint_E2_impl(x, result, 1);
}

int gsl_sf_expint_En_e(const int n, const double x, gsl_sf_result * result)
{
  return expint_En_impl(n, x, result, 0);
}


int gsl_sf_expint_En_scaled_e(const int n, const double x, gsl_sf_result * result)
{
  return expint_En_impl(n, x, result, 1);
}


int gsl_sf_expint_Ei_e(const double x, gsl_sf_result * result)
{
  /* CHECK_POINTER(result) */

  {
    int status = gsl_sf_expint_E1_e(-x, result);
    result->val = -result->val;
    return status;
  }
}


int gsl_sf_expint_Ei_scaled_e(const double x, gsl_sf_result * result)
{
  /* CHECK_POINTER(result) */

  {
    int status = gsl_sf_expint_E1_scaled_e(-x, result);
    result->val = -result->val;
    return status;
  }
}


#if 0
static double recurse_En(int n, double x, double E1)
{
  int i;
  double En = E1;
  double ex = exp(-x);
  for(i=2; i<=n; i++) {
    En = 1./(i-1) * (ex - x * En);
  }
  return En;
}
#endif


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

#include "eval.h"

double gsl_sf_expint_E1(const double x)
{
  EVAL_RESULT(gsl_sf_expint_E1_e(x, &result));
}

double gsl_sf_expint_E1_scaled(const double x)
{
  EVAL_RESULT(gsl_sf_expint_E1_scaled_e(x, &result));
}

double gsl_sf_expint_E2(const double x)
{
  EVAL_RESULT(gsl_sf_expint_E2_e(x, &result));
}

double gsl_sf_expint_E2_scaled(const double x)
{
  EVAL_RESULT(gsl_sf_expint_E2_scaled_e(x, &result));
}

double gsl_sf_expint_En(const int n, const double x)
{
  EVAL_RESULT(gsl_sf_expint_En_e(n, x, &result));
}

double gsl_sf_expint_En_scaled(const int n, const double x)
{
  EVAL_RESULT(gsl_sf_expint_En_scaled_e(n, x, &result));
}

double gsl_sf_expint_Ei(const double x)
{
  EVAL_RESULT(gsl_sf_expint_Ei_e(x, &result));
}

double gsl_sf_expint_Ei_scaled(const double x)
{
  EVAL_RESULT(gsl_sf_expint_Ei_scaled_e(x, &result));
}

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