/* sum/levin_u.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000, 2007, 2009 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 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 <gsl/gsl_math.h>
#include <gsl/gsl_test.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_sum.h>
int
gsl_sum_levin_u_accel (const double *array, const size_t array_size,
gsl_sum_levin_u_workspace * w,
double *sum_accel, double *abserr)
{
return gsl_sum_levin_u_minmax (array, array_size,
0, array_size - 1, w, sum_accel, abserr);
}
int
gsl_sum_levin_u_minmax (const double *array, const size_t array_size,
const size_t min_terms, const size_t max_terms,
gsl_sum_levin_u_workspace * w,
double *sum_accel, double *abserr)
{
/* Ignore any trailing zeros in the array */
size_t size = array_size;
while (size > 0 && array[size - 1] == 0) {
size--;
}
if (size == 0)
{
*sum_accel = 0.0;
*abserr = 0.0;
w->sum_plain = 0.0;
w->terms_used = 0;
return GSL_SUCCESS;
}
else if (size == 1)
{
*sum_accel = array[0];
*abserr = 0.0;
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 noise_n = 0.0, noise_nm1 = 0.0;
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;
double variance = 0;
size_t n;
unsigned int i;
int better = 0;
int before = 0;
int converging = 0;
double least_trunc = GSL_DBL_MAX;
double least_trunc_noise = GSL_DBL_MAX;
double least_trunc_result;
/* 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_u_step (t, n, nmax, w, &result_n);
}
least_trunc_result = result_n;
variance = 0;
for (i = 0; i < n; i++)
{
double dn = w->dsum[i] * GSL_MACH_EPS * array[i];
variance += dn * dn;
}
noise_n = sqrt (variance);
/* 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_u_step (t, n, nmax, 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);
noise_nm1 = noise_n;
variance = 0;
for (i = 0; i <= n; i++)
{
double dn = w->dsum[i] * GSL_MACH_EPS * array[i];
variance += dn * dn;
}
noise_n = sqrt (variance);
/* 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_result = result_n;
least_trunc = trunc_n;
least_trunc_noise = noise_n;
}
if (noise_n > trunc_n / 3.0)
break;
if (trunc_n < 10.0 * GSL_MACH_EPS * fabs (result_n))
break;
}
}
if (converging)
{
/* Stopped in the convergence region. Return result and
error estimate. */
*sum_accel = least_trunc_result;
*abserr = GSL_MAX_DBL (least_trunc, least_trunc_noise);
w->terms_used = n;
return GSL_SUCCESS;
}
else
{
/* Never reached the convergence region. Use the last
calculated values. */
*sum_accel = result_n;
*abserr = GSL_MAX_DBL (trunc_n, noise_n);
w->terms_used = n;
return GSL_SUCCESS;
}
}
}
int
gsl_sum_levin_u_step (const double term, const size_t n, const size_t nmax,
gsl_sum_levin_u_workspace * w, double *sum_accel)
{
#define I(i,j) ((i)*(nmax+1) + (j))
if (n == 0)
{
*sum_accel = term;
w->sum_plain = term;
w->q_den[0] = 1.0 / term;
w->q_num[0] = 1.0;
w->dq_den[I (0, 0)] = -1.0 / (term * term);
w->dq_num[I (0, 0)] = 0.0;
w->dsum[0] = 1.0;
return GSL_SUCCESS;
}
else
{
double result;
double factor = 1.0;
double ratio = (double) n / (n + 1.0);
unsigned int i;
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 (i = 0; i < n; i++)
{
w->dq_den[I (i, n)] = 0;
w->dq_num[I (i, n)] = w->q_den[n];
}
w->dq_den[I (n, n)] = -w->q_den[n] / term;
w->dq_num[I (n, n)] =
w->q_den[n] + w->sum_plain * (w->dq_den[I (n, 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];
for (i = 0; i < n; i++)
{
w->dq_den[I (i, j)] =
w->dq_den[I (i, j + 1)] - c * w->dq_den[I (i, j)];
w->dq_num[I (i, j)] =
w->dq_num[I (i, j + 1)] - c * w->dq_num[I (i, j)];
}
w->dq_den[I (n, j)] = w->dq_den[I (n, j + 1)];
w->dq_num[I (n, j)] = w->dq_num[I (n, j + 1)];
}
result = w->q_num[0] / w->q_den[0];
*sum_accel = result;
for (i = 0; i <= n; i++)
{
w->dsum[i] =
(w->dq_num[I (i, 0)] -
result * w->dq_den[I (i, 0)]) / w->q_den[0];
}
return GSL_SUCCESS;
}
}