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/* 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., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, 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;
}
}
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