levin_utrunc.c

开放gsl矩阵运算

/* sum/levin_utrunc.c
 * 
 * Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman, 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.
 */

/* Author:  G. Jungman */

#include <config.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_test.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_sum.h>

int
gsl_sum_levin_utrunc_accel (const double *array,
                            const size_t array_size,
                            gsl_sum_levin_utrunc_workspace * w,
                            double *sum_accel, double *abserr_trunc)
{
  return gsl_sum_levin_utrunc_minmax (array, array_size,
				      0, array_size - 1,
				      w, sum_accel, abserr_trunc);
}


int
gsl_sum_levin_utrunc_minmax (const double *array,
			     const size_t array_size,
			     const size_t min_terms,
			     const size_t max_terms,
			     gsl_sum_levin_utrunc_workspace * w,
			     double *sum_accel, double *abserr_trunc)
{
  if (array_size == 0)
    {
      *sum_accel = 0.0;
      *abserr_trunc = 0.0;
      w->sum_plain = 0.0;
      w->terms_used = 0;
      return GSL_SUCCESS;
    }
  else if (array_size == 1)
    {
      *sum_accel = array[0];
      *abserr_trunc = GSL_POSINF;
      w->sum_plain = array[0];
      w->terms_used = 1;
      return GSL_SUCCESS;
    }
  else
    {
      const double SMALL = 0.01;
      const size_t nmax = GSL_MAX (max_terms, array_size) - 1;
      double trunc_n = 0.0, trunc_nm1 = 0.0;
      double actual_trunc_n = 0.0, actual_trunc_nm1 = 0.0;
      double result_n = 0.0, result_nm1 = 0.0;
      size_t n;
      int better = 0;
      int before = 0;
      int converging = 0;
      double least_trunc = GSL_DBL_MAX;
      double result_least_trunc;

      /* Calculate specified minimum number of terms. No convergence
         tests are made, and no truncation information is stored. */

      for (n = 0; n < min_terms; n++)
	{
	  const double t = array[n];

	  result_nm1 = result_n;
	  gsl_sum_levin_utrunc_step (t, n, w, &result_n);
	}

      /* Assume the result after the minimum calculation is the best. */

      result_least_trunc = result_n;

      /* Calculate up to maximum number of terms. Check truncation
         condition. */

      for (; n <= nmax; n++)
	{
	  const double t = array[n];

	  result_nm1 = result_n;
	  gsl_sum_levin_utrunc_step (t, n, w, &result_n);

	  /* Compute the truncation error directly */

	  actual_trunc_nm1 = actual_trunc_n;
	  actual_trunc_n = fabs (result_n - result_nm1);

	  /* Average results to make a more reliable estimate of the
	     real truncation error */

	  trunc_nm1 = trunc_n;
	  trunc_n = 0.5 * (actual_trunc_n + actual_trunc_nm1);

	  /* Determine if we are in the convergence region. */

	  better = (trunc_n < trunc_nm1 || trunc_n < SMALL * fabs (result_n));
	  converging = converging || (better && before);
	  before = better;

	  if (converging)
	    {
	      if (trunc_n < least_trunc)
		{
		  /* Found a low truncation point in the convergence
		     region. Save it. */

		  least_trunc = trunc_n;
		  result_least_trunc = result_n;
		}

	      if (fabs (trunc_n / result_n) < 10.0 * GSL_MACH_EPS)
		break;
	    }
	}

      if (converging)
	{
	  /* Stopped in the convergence region. Return result and
	     error estimate. */

	  *sum_accel = result_least_trunc;
	  *abserr_trunc = least_trunc;
	  w->terms_used = n;
	  return GSL_SUCCESS;
	}
      else
	{
	  /* Never reached the convergence region. Use the last
	     calculated values. */

	  *sum_accel = result_n;
	  *abserr_trunc = trunc_n;
	  w->terms_used = n;
	  return GSL_SUCCESS;
	}
    }
}

int
gsl_sum_levin_utrunc_step (const double term,
			   const size_t n,
			   gsl_sum_levin_utrunc_workspace * w, double *sum_accel)
{
  if (term == 0.0)
    {
      /* This is actually harmless when treated in this way. A term
         which is exactly zero is simply ignored; the state is not
         changed. We return GSL_EZERODIV as an indicator that this
         occured. */

      return GSL_EZERODIV;
    }
  else if (n == 0)
    {
      *sum_accel = term;
      w->sum_plain = term;
      w->q_den[0] = 1.0 / term;
      w->q_num[0] = 1.0;
      return GSL_SUCCESS;
    }
  else
    {
      double factor = 1.0;
      double ratio = (double) n / (n + 1.0);
      int j;

      w->sum_plain += term;
      w->q_den[n] = 1.0 / (term * (n + 1.0) * (n + 1.0));
      w->q_num[n] = w->sum_plain * w->q_den[n];

      for (j = n - 1; j >= 0; j--)
	{
	  double c = factor * (j + 1) / (n + 1);
	  factor *= ratio;
	  w->q_den[j] = w->q_den[j + 1] - c * w->q_den[j];
	  w->q_num[j] = w->q_num[j + 1] - c * w->q_num[j];
	}

      *sum_accel = w->q_num[0] / w->q_den[0];
      return GSL_SUCCESS;
    }
}