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/* roots/steffenson.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000 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 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.
*/
/* 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 <config.h>
#include <stddef.h>
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include <float.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_roots.h>
#include "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 (!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 (!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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