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qags.c

/* integration/qags.c
 * 
 * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2007 Brian Gough
 * 
 * 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.
 */

#include <config.h>
#include <stdlib.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_integration.h>

#include "initialise.c"
#include "set_initial.c"
#include "qpsrt.c"
#include "util.c"
#include "reset.c"
#include "qpsrt2.c"
#include "qelg.c"
#include "positivity.c"

static int qags (const gsl_function * f, const double a, const double
  b, const double epsabs, const double epsrel, const size_t limit,
  gsl_integration_workspace * workspace, double *result, double *abserr,
  gsl_integration_rule * q);

int
gsl_integration_qags (const gsl_function *f,
                      double a, double b,
                      double epsabs, double epsrel, size_t limit,
                      gsl_integration_workspace * workspace,
                      double * result, double * abserr)
{
  int status = qags (f, a, b, epsabs, epsrel, limit,
                     workspace, 
                     result, abserr, 
                     &gsl_integration_qk21) ;
  return status ;
}

/* QAGI: evaluate an integral over an infinite range using the
   transformation

   integrate(f(x),-Inf,Inf) = integrate((f((1-t)/t) + f(-(1-t)/t))/t^2,0,1)

   */

static double i_transform (double t, void *params);

int
gsl_integration_qagi (gsl_function * f,
                      double epsabs, double epsrel, size_t limit,
                      gsl_integration_workspace * workspace,
                      double *result, double *abserr)
{
  int status;

  gsl_function f_transform;

  f_transform.function = &i_transform;
  f_transform.params = f;

  status = qags (&f_transform, 0.0, 1.0, 
                 epsabs, epsrel, limit,
                 workspace,
                 result, abserr,
                 &gsl_integration_qk15);

  return status;
}

static double 
i_transform (double t, void *params)
{
  gsl_function *f = (gsl_function *) params;
  double x = (1 - t) / t;
  double y = GSL_FN_EVAL (f, x) + GSL_FN_EVAL (f, -x);
  return (y / t) / t;
}


/* QAGIL: Evaluate an integral over an infinite range using the
   transformation,
   
   integrate(f(x),-Inf,b) = integrate(f(b-(1-t)/t)/t^2,0,1)

   */

struct il_params { double b ; gsl_function * f ; } ;

static double il_transform (double t, void *params);

int
gsl_integration_qagil (gsl_function * f,
                       double b,
                       double epsabs, double epsrel, size_t limit,
                       gsl_integration_workspace * workspace,
                       double *result, double *abserr)
{
  int status;

  gsl_function f_transform;
  struct il_params transform_params  ;

  transform_params.b = b ;
  transform_params.f = f ;

  f_transform.function = &il_transform;
  f_transform.params = &transform_params;

  status = qags (&f_transform, 0.0, 1.0, 
                 epsabs, epsrel, limit,
                 workspace,
                 result, abserr,
                 &gsl_integration_qk15);

  return status;
}

static double 
il_transform (double t, void *params)
{
  struct il_params *p = (struct il_params *) params;
  double b = p->b;
  gsl_function * f = p->f;
  double x = b - (1 - t) / t;
  double y = GSL_FN_EVAL (f, x);
  return (y / t) / t;
}

/* QAGIU: Evaluate an integral over an infinite range using the
   transformation

   integrate(f(x),a,Inf) = integrate(f(a+(1-t)/t)/t^2,0,1)

   */

struct iu_params { double a ; gsl_function * f ; } ;

static double iu_transform (double t, void *params);

int
gsl_integration_qagiu (gsl_function * f,
                       double a,
                       double epsabs, double epsrel, size_t limit,
                       gsl_integration_workspace * workspace,
                       double *result, double *abserr)
{
  int status;

  gsl_function f_transform;
  struct iu_params transform_params  ;

  transform_params.a = a ;
  transform_params.f = f ;

  f_transform.function = &iu_transform;
  f_transform.params = &transform_params;

  status = qags (&f_transform, 0.0, 1.0, 
                 epsabs, epsrel, limit,
                 workspace,
                 result, abserr,
                 &gsl_integration_qk15);

  return status;
}

static double 
iu_transform (double t, void *params)
{
  struct iu_params *p = (struct iu_params *) params;
  double a = p->a;
  gsl_function * f = p->f;
  double x = a + (1 - t) / t;
  double y = GSL_FN_EVAL (f, x);
  return (y / t) / t;
}

/* Main integration function */

static int
qags (const gsl_function * f,
      const double a, const double b,
      const double epsabs, const double epsrel,
      const size_t limit,
      gsl_integration_workspace * workspace,
      double *result, double *abserr,
      gsl_integration_rule * q)
{
  double area, errsum;
  double res_ext, err_ext;
  double result0, abserr0, resabs0, resasc0;
  double tolerance;

  double ertest = 0;
  double error_over_large_intervals = 0;
  double reseps = 0, abseps = 0, correc = 0;
  size_t ktmin = 0;
  int roundoff_type1 = 0, roundoff_type2 = 0, roundoff_type3 = 0;
  int error_type = 0, error_type2 = 0;

  size_t iteration = 0;

  int positive_integrand = 0;
  int extrapolate = 0;
  int disallow_extrapolation = 0;

  struct extrapolation_table table;

  /* Initialize results */

  initialise (workspace, a, b);

  *result = 0;
  *abserr = 0;

  if (limit > workspace->limit)
    {
      GSL_ERROR ("iteration limit exceeds available workspace", GSL_EINVAL) ;
    }

  /* Test on accuracy */

  if (epsabs <= 0 && (epsrel < 50 * GSL_DBL_EPSILON || epsrel < 0.5e-28))
    {
      GSL_ERROR ("tolerance cannot be acheived with given epsabs and epsrel",
                 GSL_EBADTOL);
    }

  /* Perform the first integration */

  q (f, a, b, &result0, &abserr0, &resabs0, &resasc0);

  set_initial_result (workspace, result0, abserr0);

  tolerance = GSL_MAX_DBL (epsabs, epsrel * fabs (result0));

  if (abserr0 <= 100 * GSL_DBL_EPSILON * resabs0 && abserr0 > tolerance)
    {
      *result = result0;
      *abserr = abserr0;

      GSL_ERROR ("cannot reach tolerance because of roundoff error"
                 "on first attempt", GSL_EROUND);
    }
  else if ((abserr0 <= tolerance && abserr0 != resasc0) || abserr0 == 0.0)
    {
      *result = result0;
      *abserr = abserr0;

      return GSL_SUCCESS;
    }
  else if (limit == 1)
    {
      *result = result0;
      *abserr = abserr0;

      GSL_ERROR ("a maximum of one iteration was insufficient", GSL_EMAXITER);
    }

  /* Initialization */

  initialise_table (&table);
  append_table (&table, result0);

  area = result0;
  errsum = abserr0;

  res_ext = result0;
  err_ext = GSL_DBL_MAX;

  positive_integrand = test_positivity (result0, resabs0);

  iteration = 1;

  do
    {
      size_t current_level;
      double a1, b1, a2, b2;
      double a_i, b_i, r_i, e_i;
      double area1 = 0, area2 = 0, area12 = 0;
      double error1 = 0, error2 = 0, error12 = 0;
      double resasc1, resasc2;
      double resabs1, resabs2;
      double last_e_i;

      /* Bisect the subinterval with the largest error estimate */

      retrieve (workspace, &a_i, &b_i, &r_i, &e_i);

      current_level = workspace->level[workspace->i] + 1;

      a1 = a_i;
      b1 = 0.5 * (a_i + b_i);
      a2 = b1;
      b2 = b_i;

      iteration++;

      q (f, a1, b1, &area1, &error1, &resabs1, &resasc1);
      q (f, a2, b2, &area2, &error2, &resabs2, &resasc2);

      area12 = area1 + area2;
      error12 = error1 + error2;
      last_e_i = e_i;

      /* Improve previous approximations to the integral and test for
         accuracy.

         We write these expressions in the same way as the original
         QUADPACK code so that the rounding errors are the same, which
         makes testing easier. */

      errsum = errsum + error12 - e_i;
      area = area + area12 - r_i;

      tolerance = GSL_MAX_DBL (epsabs, epsrel * fabs (area));

      if (resasc1 != error1 && resasc2 != error2)
        {
          double delta = r_i - area12;

          if (fabs (delta) <= 1.0e-5 * fabs (area12) && error12 >= 0.99 * e_i)
            {
              if (!extrapolate)
                {
                  roundoff_type1++;
                }
              else
                {
                  roundoff_type2++;
                }
            }
          if (iteration > 10 && error12 > e_i)
            {
              roundoff_type3++;
            }
        }

      /* Test for roundoff and eventually set error flag */

      if (roundoff_type1 + roundoff_type2 >= 10 || roundoff_type3 >= 20)
        {
          error_type = 2;       /* round off error */
        }

      if (roundoff_type2 >= 5)
        {
          error_type2 = 1;
        }

      /* set error flag in the case of bad integrand behaviour at
         a point of the integration range */

      if (subinterval_too_small (a1, a2, b2))
        {
          error_type = 4;
        }

      /* append the newly-created intervals to the list */

      update (workspace, a1, b1, area1, error1, a2, b2, area2, error2);

      if (errsum <= tolerance)
        {
          goto compute_result;
        }

      if (error_type)
        {
          break;
        }

      if (iteration >= limit - 1)
        {
          error_type = 1;
          break;
        }

      if (iteration == 2)       /* set up variables on first iteration */
        {
          error_over_large_intervals = errsum;
          ertest = tolerance;
          append_table (&table, area);
          continue;
        }

      if (disallow_extrapolation)
        {
          continue;
        }

      error_over_large_intervals += -last_e_i;

      if (current_level < workspace->maximum_level)
        {
          error_over_large_intervals += error12;
        }

      if (!extrapolate)
        {
          /* test whether the interval to be bisected next is the
             smallest interval. */

          if (large_interval (workspace))
            continue;

          extrapolate = 1;
          workspace->nrmax = 1;
        }

      if (!error_type2 && error_over_large_intervals > ertest)
        {
          if (increase_nrmax (workspace))
            continue;
        }

      /* Perform extrapolation */

      append_table (&table, area);

      qelg (&table, &reseps, &abseps);

      ktmin++;

      if (ktmin > 5 && err_ext < 0.001 * errsum)
        {
          error_type = 5;
        }

      if (abseps < err_ext)
        {
          ktmin = 0;
          err_ext = abseps;
          res_ext = reseps;
          correc = error_over_large_intervals;
          ertest = GSL_MAX_DBL (epsabs, epsrel * fabs (reseps));
          if (err_ext <= ertest)
            break;
        }

      /* Prepare bisection of the smallest interval. */

      if (table.n == 1)
        {
          disallow_extrapolation = 1;
        }

      if (error_type == 5)
        {
          break;
        }

      /* work on interval with largest error */

      reset_nrmax (workspace);
      extrapolate = 0;
      error_over_large_intervals = errsum;

    }
  while (iteration < limit);

  *result = res_ext;
  *abserr = err_ext;

  if (err_ext == GSL_DBL_MAX)
    goto compute_result;

  if (error_type || error_type2)
    {
      if (error_type2)
        {
          err_ext += correc;
        }

      if (error_type == 0)
        error_type = 3;

      if (res_ext != 0.0 && area != 0.0)
        {
          if (err_ext / fabs (res_ext) > errsum / fabs (area))
            goto compute_result;
        }
      else if (err_ext > errsum)
        {
          goto compute_result;
        }
      else if (area == 0.0)
        {
          goto return_error;
        }
    }

  /*  Test on divergence. */

  {
    double max_area = GSL_MAX_DBL (fabs (res_ext), fabs (area));

    if (!positive_integrand && max_area < 0.01 * resabs0)
      goto return_error;
  }

  {
    double ratio = res_ext / area;

    if (ratio < 0.01 || ratio > 100.0 || errsum > fabs (area))
      error_type = 6;
  }

  goto return_error;

compute_result:

  *result = sum_results (workspace);
  *abserr = errsum;

return_error:

  if (error_type > 2)
    error_type--;



  if (error_type == 0) 
    {
      return GSL_SUCCESS;
    }
  else if (error_type == 1)
    {
      GSL_ERROR ("number of iterations was insufficient", GSL_EMAXITER);
    }
  else if (error_type == 2)
    {
      GSL_ERROR ("cannot reach tolerance because of roundoff error",
                 GSL_EROUND);
    }
  else if (error_type == 3)
    {
      GSL_ERROR ("bad integrand behavior found in the integration interval",
                 GSL_ESING);
    }
  else if (error_type == 4)
    {
      GSL_ERROR ("roundoff error detected in the extrapolation table",
                 GSL_EROUND);
    }
  else if (error_type == 5)
    {
      GSL_ERROR ("integral is divergent, or slowly convergent",
                 GSL_EDIVERGE);
    }
  else
    {
      GSL_ERROR ("could not integrate function", GSL_EFAILED);
    }

}

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