| /* 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; |