/* 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;