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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 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 "gsl__config.h"
- #include <stdlib.h>
- #include "gsl_math.h"
- #include "gsl_errno.h"
- #include "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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