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broyden.c

/* multiroots/broyden.c
 * 
 * Copyright (C) 1996, 1997, 1998, 1999, 2000 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.
 */

#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_multiroots.h>
#include <gsl/gsl_linalg.h>

#include "enorm.c"

/* Broyden's method. It is not an efficient or modern algorithm but
   gives an example of a rank-1 update.

   C.G. Broyden, "A Class of Methods for Solving Nonlinear
   Simultaneous Equations", Mathematics of Computation, vol 19 (1965),
   p 577-593

 */

typedef struct
  {
    gsl_matrix *H;
    gsl_matrix *lu;
    gsl_permutation *permutation;
    gsl_vector *v;
    gsl_vector *w;
    gsl_vector *y;
    gsl_vector *p;
    gsl_vector *fnew;
    gsl_vector *x_trial;
    double phi;
  }
broyden_state_t;

static int broyden_alloc (void *vstate, size_t n);
static int broyden_set (void *vstate, gsl_multiroot_function * function, gsl_vector * x, gsl_vector * f, gsl_vector * dx);
static int broyden_iterate (void *vstate, gsl_multiroot_function * function, gsl_vector * x, gsl_vector * f, gsl_vector * dx);
static void broyden_free (void *vstate);


static int
broyden_alloc (void *vstate, size_t n)
{
  broyden_state_t *state = (broyden_state_t *) vstate;
  gsl_vector *v, *w, *y, *fnew, *x_trial, *p;
  gsl_permutation *perm;
  gsl_matrix *m, *H;

  m = gsl_matrix_calloc (n, n);

  if (m == 0)
    {
      GSL_ERROR ("failed to allocate space for lu", GSL_ENOMEM);
    }

  state->lu = m;

  perm = gsl_permutation_calloc (n);

  if (perm == 0)
    {
      gsl_matrix_free (m);

      GSL_ERROR ("failed to allocate space for permutation", GSL_ENOMEM);
    }

  state->permutation = perm;

  H = gsl_matrix_calloc (n, n);

  if (H == 0)
    {
      gsl_permutation_free (perm);
      gsl_matrix_free (m);

      GSL_ERROR ("failed to allocate space for d", GSL_ENOMEM);
    }

  state->H = H;

  v = gsl_vector_calloc (n);

  if (v == 0)
    {
      gsl_matrix_free (H);
      gsl_permutation_free (perm);
      gsl_matrix_free (m);

      GSL_ERROR ("failed to allocate space for v", GSL_ENOMEM);
    }

  state->v = v;

  w = gsl_vector_calloc (n);

  if (w == 0)
    {
      gsl_vector_free (v);
      gsl_matrix_free (H);
      gsl_permutation_free (perm);
      gsl_matrix_free (m);

      GSL_ERROR ("failed to allocate space for w", GSL_ENOMEM);
    }

  state->w = w;

  y = gsl_vector_calloc (n);

  if (y == 0)
    {
      gsl_vector_free (w);
      gsl_vector_free (v);
      gsl_matrix_free (H);
      gsl_permutation_free (perm);
      gsl_matrix_free (m);

      GSL_ERROR ("failed to allocate space for y", GSL_ENOMEM);
    }

  state->y = y;

  fnew = gsl_vector_calloc (n);

  if (fnew == 0)
    {
      gsl_vector_free (y);
      gsl_vector_free (w);
      gsl_vector_free (v);
      gsl_matrix_free (H);
      gsl_permutation_free (perm);
      gsl_matrix_free (m);

      GSL_ERROR ("failed to allocate space for fnew", GSL_ENOMEM);
    }

  state->fnew = fnew;

  x_trial = gsl_vector_calloc (n);

  if (x_trial == 0)
    {
      gsl_vector_free (fnew);
      gsl_vector_free (y);
      gsl_vector_free (w);
      gsl_vector_free (v);
      gsl_matrix_free (H);
      gsl_permutation_free (perm);
      gsl_matrix_free (m);

      GSL_ERROR ("failed to allocate space for x_trial", GSL_ENOMEM);
    }

  state->x_trial = x_trial;

  p = gsl_vector_calloc (n);

  if (p == 0)
    {
      gsl_vector_free (x_trial);
      gsl_vector_free (fnew);
      gsl_vector_free (y);
      gsl_vector_free (w);
      gsl_vector_free (v);
      gsl_matrix_free (H);
      gsl_permutation_free (perm);
      gsl_matrix_free (m);

      GSL_ERROR ("failed to allocate space for p", GSL_ENOMEM);
    }

  state->p = p;

  return GSL_SUCCESS;
}

static int
broyden_set (void *vstate, gsl_multiroot_function * function, gsl_vector * x, gsl_vector * f, gsl_vector * dx)
{
  broyden_state_t *state = (broyden_state_t *) vstate;
  size_t i, j, n = function->n;
  int signum = 0;

  GSL_MULTIROOT_FN_EVAL (function, x, f);

  gsl_multiroot_fdjacobian (function, x, f, GSL_SQRT_DBL_EPSILON, state->lu);
  gsl_linalg_LU_decomp (state->lu, state->permutation, &signum);
  gsl_linalg_LU_invert (state->lu, state->permutation, state->H);

  for (i = 0; i < n; i++)
    for (j = 0; j < n; j++)
      gsl_matrix_set(state->H,i,j,-gsl_matrix_get(state->H,i,j));

  for (i = 0; i < n; i++)
    {
      gsl_vector_set (dx, i, 0.0);
    }

  state->phi = enorm (f);

  return GSL_SUCCESS;
}

static int
broyden_iterate (void *vstate, gsl_multiroot_function * function, gsl_vector * x, gsl_vector * f, gsl_vector * dx)
{
  broyden_state_t *state = (broyden_state_t *) vstate;

  double phi0, phi1, t, lambda;

  gsl_matrix *H = state->H;
  gsl_vector *p = state->p;
  gsl_vector *y = state->y;
  gsl_vector *v = state->v;
  gsl_vector *w = state->w;
  gsl_vector *fnew = state->fnew;
  gsl_vector *x_trial = state->x_trial;
  gsl_matrix *lu = state->lu;
  gsl_permutation *perm = state->permutation;

  size_t i, j, iter;

  size_t n = function->n;

  /* p = H f */

  for (i = 0; i < n; i++)
    {
      double sum = 0;

      for (j = 0; j < n; j++)
        {
          sum += gsl_matrix_get (H, i, j) * gsl_vector_get (f, j);
        }
      gsl_vector_set (p, i, sum);
    }

  t = 1;
  iter = 0;

  phi0 = state->phi;

new_step:

  for (i = 0; i < n; i++)
    {
      double pi = gsl_vector_get (p, i);
      double xi = gsl_vector_get (x, i);
      gsl_vector_set (x_trial, i, xi + t * pi);
    }

  { 
    int status = GSL_MULTIROOT_FN_EVAL (function, x_trial, fnew);

    if (status != GSL_SUCCESS) 
      {
        return GSL_EBADFUNC;
      }
  }

  phi1 = enorm (fnew);

  iter++ ;

  if (phi1 > phi0 && iter < 10 && t > 0.1)
    {
      /* full step goes uphill, take a reduced step instead */
      
      double theta = phi1 / phi0;
      t *= (sqrt (1.0 + 6.0 * theta) - 1.0) / (3.0 * theta);
      goto new_step;
    }

  if (phi1 > phi0)
    {
      /* need to recompute Jacobian */
      int signum = 0;
      
      gsl_multiroot_fdjacobian (function, x, f, GSL_SQRT_DBL_EPSILON, lu);
      
      for (i = 0; i < n; i++)
        for (j = 0; j < n; j++)
          gsl_matrix_set(lu,i,j,-gsl_matrix_get(lu,i,j));
      
      gsl_linalg_LU_decomp (lu, perm, &signum);
      gsl_linalg_LU_invert (lu, perm, H);
      
      gsl_linalg_LU_solve (lu, perm, f, p);          

      t = 1;

      for (i = 0; i < n; i++)
        {
          double pi = gsl_vector_get (p, i);
          double xi = gsl_vector_get (x, i);
          gsl_vector_set (x_trial, i, xi + t * pi);
        }
      
      {
        int status = GSL_MULTIROOT_FN_EVAL (function, x_trial, fnew);
        
        if (status != GSL_SUCCESS) 
          {
            return GSL_EBADFUNC;
          }
      }
      
      phi1 = enorm (fnew);
    }
  
  /* y = f' - f */

  for (i = 0; i < n; i++)
    {
      double yi = gsl_vector_get (fnew, i) - gsl_vector_get (f, i);
      gsl_vector_set (y, i, yi);
    }

  /* v = H y */

  for (i = 0; i < n; i++)
    {
      double sum = 0;

      for (j = 0; j < n; j++)
        {
          sum += gsl_matrix_get (H, i, j) * gsl_vector_get (y, j);
        }

      gsl_vector_set (v, i, sum);
    }

  /* lambda = p . v */

  lambda = 0;

  for (i = 0; i < n; i++)
    {
      lambda += gsl_vector_get (p, i) * gsl_vector_get (v, i);
    }

  if (lambda == 0)
    {
      GSL_ERROR ("approximation to Jacobian has collapsed", GSL_EZERODIV) ;
    }

  /* v' = v + t * p */

  for (i = 0; i < n; i++)
    {
      double vi = gsl_vector_get (v, i) + t * gsl_vector_get (p, i);
      gsl_vector_set (v, i, vi);
    }

  /* w^T = p^T H */

  for (i = 0; i < n; i++)
    {
      double sum = 0;

      for (j = 0; j < n; j++)
        {
          sum += gsl_matrix_get (H, j, i) * gsl_vector_get (p, j);
        }

      gsl_vector_set (w, i, sum);
    }

  /* Hij -> Hij - (vi wj / lambda) */

  for (i = 0; i < n; i++)
    {
      double vi = gsl_vector_get (v, i);

      for (j = 0; j < n; j++)
        {
          double wj = gsl_vector_get (w, j);
          double Hij = gsl_matrix_get (H, i, j) - vi * wj / lambda;
          gsl_matrix_set (H, i, j, Hij);
        }
    }

  /* copy fnew into f */

  gsl_vector_memcpy (f, fnew);

  /* copy x_trial into x */

  gsl_vector_memcpy (x, x_trial);

  for (i = 0; i < n; i++)
    {
      double pi = gsl_vector_get (p, i);
      gsl_vector_set (dx, i, t * pi);
    }

  state->phi = phi1;

  return GSL_SUCCESS;
}


static void
broyden_free (void *vstate)
{
  broyden_state_t *state = (broyden_state_t *) vstate;

  gsl_matrix_free (state->H);

  gsl_matrix_free (state->lu);
  gsl_permutation_free (state->permutation);
  
  gsl_vector_free (state->v);
  gsl_vector_free (state->w);
  gsl_vector_free (state->y);
  gsl_vector_free (state->p);

  gsl_vector_free (state->fnew);
  gsl_vector_free (state->x_trial);
  
}


static const gsl_multiroot_fsolver_type broyden_type =
{"broyden",                     /* name */
 sizeof (broyden_state_t),
 &broyden_alloc,
 &broyden_set,
 &broyden_iterate,
 &broyden_free};

const gsl_multiroot_fsolver_type *gsl_multiroot_fsolver_broyden = &broyden_type;

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