1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
|
/* 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 <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 (!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;
|
