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/* diff/diff.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000 David Morrison
*
* 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.
*/
#include <config.h>
#include <stdlib.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_diff.h>
int
gsl_diff_backward (const gsl_function * f,
double x, double *result, double *abserr)
{
/* Construct a divided difference table with a fairly large step
size to get a very rough estimate of f''. Use this to estimate
the step size which will minimize the error in calculating f'. */
int i, k;
double h = GSL_SQRT_DBL_EPSILON;
double a[3], d[3], a2;
/* Algorithm based on description on pg. 204 of Conte and de Boor
(CdB) - coefficients of Newton form of polynomial of degree 2. */
for (i = 0; i < 3; i++)
{
a[i] = x + (i - 2.0) * h;
d[i] = GSL_FN_EVAL (f, a[i]);
}
for (k = 1; k < 4; k++)
{
for (i = 0; i < 3 - k; i++)
{
d[i] = (d[i + 1] - d[i]) / (a[i + k] - a[i]);
}
}
/* Adapt procedure described on pg. 282 of CdB to find best value of
step size. */
a2 = fabs (d[0] + d[1] + d[2]);
if (a2 < 100.0 * GSL_SQRT_DBL_EPSILON)
{
a2 = 100.0 * GSL_SQRT_DBL_EPSILON;
}
h = sqrt (GSL_SQRT_DBL_EPSILON / (2.0 * a2));
if (h > 100.0 * GSL_SQRT_DBL_EPSILON)
{
h = 100.0 * GSL_SQRT_DBL_EPSILON;
}
*result = (GSL_FN_EVAL (f, x) - GSL_FN_EVAL (f, x - h)) / h;
*abserr = fabs (10.0 * a2 * h);
return GSL_SUCCESS;
}
int
gsl_diff_forward (const gsl_function * f,
double x, double *result, double *abserr)
{
/* Construct a divided difference table with a fairly large step
size to get a very rough estimate of f''. Use this to estimate
the step size which will minimize the error in calculating f'. */
int i, k;
double h = GSL_SQRT_DBL_EPSILON;
double a[3], d[3], a2;
/* Algorithm based on description on pg. 204 of Conte and de Boor
(CdB) - coefficients of Newton form of polynomial of degree 2. */
for (i = 0; i < 3; i++)
{
a[i] = x + i * h;
d[i] = GSL_FN_EVAL (f, a[i]);
}
for (k = 1; k < 4; k++)
{
for (i = 0; i < 3 - k; i++)
{
d[i] = (d[i + 1] - d[i]) / (a[i + k] - a[i]);
}
}
/* Adapt procedure described on pg. 282 of CdB to find best value of
step size. */
a2 = fabs (d[0] + d[1] + d[2]);
if (a2 < 100.0 * GSL_SQRT_DBL_EPSILON)
{
a2 = 100.0 * GSL_SQRT_DBL_EPSILON;
}
h = sqrt (GSL_SQRT_DBL_EPSILON / (2.0 * a2));
if (h > 100.0 * GSL_SQRT_DBL_EPSILON)
{
h = 100.0 * GSL_SQRT_DBL_EPSILON;
}
*result = (GSL_FN_EVAL (f, x + h) - GSL_FN_EVAL (f, x)) / h;
*abserr = fabs (10.0 * a2 * h);
return GSL_SUCCESS;
}
int
gsl_diff_central (const gsl_function * f,
double x, double *result, double *abserr)
{
/* Construct a divided difference table with a fairly large step
size to get a very rough estimate of f'''. Use this to estimate
the step size which will minimize the error in calculating f'. */
int i, k;
double h = GSL_SQRT_DBL_EPSILON;
double a[4], d[4], a3;
/* Algorithm based on description on pg. 204 of Conte and de Boor
(CdB) - coefficients of Newton form of polynomial of degree 3. */
for (i = 0; i < 4; i++)
{
a[i] = x + (i - 2.0) * h;
d[i] = GSL_FN_EVAL (f, a[i]);
}
for (k = 1; k < 5; k++)
{
for (i = 0; i < 4 - k; i++)
{
d[i] = (d[i + 1] - d[i]) / (a[i + k] - a[i]);
}
}
/* Adapt procedure described on pg. 282 of CdB to find best
value of step size. */
a3 = fabs (d[0] + d[1] + d[2] + d[3]);
if (a3 < 100.0 * GSL_SQRT_DBL_EPSILON)
{
a3 = 100.0 * GSL_SQRT_DBL_EPSILON;
}
h = pow (GSL_SQRT_DBL_EPSILON / (2.0 * a3), 1.0 / 3.0);
if (h > 100.0 * GSL_SQRT_DBL_EPSILON)
{
h = 100.0 * GSL_SQRT_DBL_EPSILON;
}
*result = (GSL_FN_EVAL (f, x + h) - GSL_FN_EVAL (f, x - h)) / (2.0 * h);
*abserr = fabs (100.0 * a3 * h * h);
return GSL_SUCCESS;
}
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