qag.c

该文件为c++的数学函数库!是一个非常有用的编程工具.它含有各种数学函数,为科学计算、工程应用等程序编写提供方便！

/* integration/qag.c
 * 
 * Copyright (C) 1996, 1997, 1998, 1999, 2000 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 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., 675 Mass Ave, Cambridge, MA 02139, USA.
 */

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

#include "initialise.c"
#include "set_initial.c"
#include "qpsrt.c"
#include "util.c"

static int
qag (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_qag (const gsl_function *f,
                     double a, double b,
                     double epsabs, double epsrel, size_t limit,
                     int key,
                     gsl_integration_workspace * workspace,
                     double * result, double * abserr)
{
  int status ;
  gsl_integration_rule * integration_rule = gsl_integration_qk15 ;

  if (key < GSL_INTEG_GAUSS15)
    {
      key = GSL_INTEG_GAUSS15 ;
    } 
  else if (key > GSL_INTEG_GAUSS61) 
    {
      key = GSL_INTEG_GAUSS61 ;
    }

  switch (key) 
    {
    case GSL_INTEG_GAUSS15:
      integration_rule = gsl_integration_qk15 ;
      break ;
    case GSL_INTEG_GAUSS21:
      integration_rule = gsl_integration_qk21 ;
      break ;
    case GSL_INTEG_GAUSS31:
      integration_rule = gsl_integration_qk31 ; 
      break ;
    case GSL_INTEG_GAUSS41:
      integration_rule = gsl_integration_qk41 ;
      break ;      
    case GSL_INTEG_GAUSS51:
      integration_rule = gsl_integration_qk51 ;
      break ;      
    case GSL_INTEG_GAUSS61:
      integration_rule = gsl_integration_qk61 ;
      break ;      
    default:
      GSL_ERROR("value of key does specify a known integration rule", 
                GSL_EINVAL) ;
    }

  status = qag (f, a, b, epsabs, epsrel, limit,
                workspace, 
                result, abserr, 
                integration_rule) ;
  
  return status ;
}

static int
qag (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 result0, abserr0, resabs0, resasc0;
  double tolerance;
  size_t iteration = 0;
  int roundoff_type1 = 0, roundoff_type2 = 0, error_type = 0;

  double round_off;     

  /* Initialize results */

  initialise (workspace, a, b);

  *result = 0;
  *abserr = 0;

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

  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);

  /* Test on accuracy */

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

  /* need IEEE rounding here to match original quadpack behavior */

  round_off = GSL_COERCE_DBL (50 * GSL_DBL_EPSILON * resabs0);

  if (abserr0 <= round_off && 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);
    }

  area = result0;
  errsum = abserr0;

  iteration = 1;

  do
    {
      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;

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

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

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

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

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

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

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

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

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

      if (errsum > tolerance)
        {
          if (roundoff_type1 >= 6 || roundoff_type2 >= 20)
            {
              error_type = 2;   /* round off error */
            }

          /* 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 = 3;
            }
        }

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

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

      iteration++;

    }
  while (iteration < limit && !error_type && errsum > tolerance);

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

  if (errsum <= tolerance)
    {
      return GSL_SUCCESS;
    }
  else if (error_type == 2)
    {
      GSL_ERROR ("roundoff error prevents tolerance from being achieved",
                 GSL_EROUND);
    }
  else if (error_type == 3)
    {
      GSL_ERROR ("bad integrand behavior found in the integration interval",
                 GSL_ESING);
    }
  else if (iteration == limit)
    {
      GSL_ERROR ("maximum number of subdivisions reached", GSL_EMAXITER);
    }
  else
    {
      GSL_ERROR ("could not integrate function", GSL_EFAILED);
    }
}