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- /* roots/steffenson.c
- *
- * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2007 Reid Priedhorsky, 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.
- */
- /* steffenson.c -- steffenson root finding algorithm
- This is Newton's method with an Aitken "delta-squared"
- acceleration of the iterates. This can improve the convergence on
- multiple roots where the ordinary Newton algorithm is slow.
- x[i+1] = x[i] - f(x[i]) / f'(x[i])
- x_accelerated[i] = x[i] - (x[i+1] - x[i])**2 / (x[i+2] - 2*x[i+1] + x[i])
- We can only use the accelerated estimate after three iterations,
- and use the unaccelerated value until then.
- */
- #include "gsl__config.h"
- #include <stddef.h>
- #include <stdlib.h>
- #include <stdio.h>
- #include <math.h>
- #include <float.h>
- #include "gsl_math.h"
- #include "gsl_errno.h"
- #include "gsl_roots.h"
- #include "gsl_roots__roots.h"
- typedef struct
- {
- double f, df;
- double x;
- double x_1;
- double x_2;
- int count;
- }
- steffenson_state_t;
- static int steffenson_init (void * vstate, gsl_function_fdf * fdf, double * root);
- static int steffenson_iterate (void * vstate, gsl_function_fdf * fdf, double * root);
- static int
- steffenson_init (void * vstate, gsl_function_fdf * fdf, double * root)
- {
- steffenson_state_t * state = (steffenson_state_t *) vstate;
- const double x = *root ;
- state->f = GSL_FN_FDF_EVAL_F (fdf, x);
- state->df = GSL_FN_FDF_EVAL_DF (fdf, x) ;
- state->x = x;
- state->x_1 = 0.0;
- state->x_2 = 0.0;
- state->count = 1;
- return GSL_SUCCESS;
- }
- static int
- steffenson_iterate (void * vstate, gsl_function_fdf * fdf, double * root)
- {
- steffenson_state_t * state = (steffenson_state_t *) vstate;
-
- double x_new, f_new, df_new;
- double x_1 = state->x_1 ;
- double x = state->x ;
- if (state->df == 0.0)
- {
- GSL_ERROR("derivative is zero", GSL_EZERODIV);
- }
- x_new = x - (state->f / state->df);
-
- GSL_FN_FDF_EVAL_F_DF(fdf, x_new, &f_new, &df_new);
- state->x_2 = x_1 ;
- state->x_1 = x ;
- state->x = x_new;
- state->f = f_new ;
- state->df = df_new ;
- if (!gsl_finite (f_new))
- {
- GSL_ERROR ("function value is not finite", GSL_EBADFUNC);
- }
- if (state->count < 3)
- {
- *root = x_new ;
- state->count++ ;
- }
- else
- {
- double u = (x - x_1) ;
- double v = (x_new - 2 * x + x_1);
- if (v == 0)
- *root = x_new; /* avoid division by zero */
- else
- *root = x_1 - u * u / v ; /* accelerated value */
- }
- if (!gsl_finite (df_new))
- {
- GSL_ERROR ("derivative value is not finite", GSL_EBADFUNC);
- }
-
- return GSL_SUCCESS;
- }
- static const gsl_root_fdfsolver_type steffenson_type =
- {"steffenson", /* name */
- sizeof (steffenson_state_t),
- &steffenson_init,
- &steffenson_iterate};
- const gsl_root_fdfsolver_type * gsl_root_fdfsolver_steffenson = &steffenson_type;
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