/* multilarge_nlinear/cholesky.c
*
* Copyright (C) 2016 Patrick Alken
*
* 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.
*/
/*
* This module calculates the solution of the normal equations least squares
* system:
*
* [ J^T J + mu D^T D ] p = -J^T f
*
* using the modified Cholesky decomposition.
*/
#include <config.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_linalg.h>
#include <gsl/gsl_multilarge_nlinear.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_permutation.h>
#include "common.c"
typedef struct
{
gsl_matrix *JTJ; /* J^T J */
gsl_matrix *work_JTJ; /* copy of J^T J */
gsl_vector *rhs; /* -J^T f, size p */
gsl_permutation *perm; /* permutation matrix for modified Cholesky */
gsl_vector *work3p; /* workspace, size 3*p */
gsl_vector *workn; /* workspace, size n */
double mu; /* current regularization parameter */
} cholesky_state_t;
static void *cholesky_alloc (const size_t n, const size_t p);
static int cholesky_init(const void * vtrust_state, void * vstate);
static int cholesky_presolve(const double mu, const void * vtrust_state, void * vstate);
static int cholesky_solve(const gsl_vector * g, gsl_vector *x,
const void * vtrust_state, void *vstate);
static int cholesky_rcond(double * rcond, const gsl_matrix * JTJ, void * vstate);
static int cholesky_covar(const gsl_matrix * JTJ, gsl_matrix * covar, void * vstate);
static int cholesky_solve_rhs(const gsl_vector * b, gsl_vector *x, cholesky_state_t *state);
static int cholesky_regularize(const double mu, const gsl_vector * diag, gsl_matrix * A,
cholesky_state_t * state);
static void *
cholesky_alloc (const size_t n, const size_t p)
{
cholesky_state_t *state;
state = calloc(1, sizeof(cholesky_state_t));
if (state == NULL)
{
GSL_ERROR_NULL ("failed to allocate cholesky state", GSL_ENOMEM);
}
state->JTJ = gsl_matrix_alloc(p, p);
if (state->JTJ == NULL)
{
GSL_ERROR_NULL ("failed to allocate space for JTJ", GSL_ENOMEM);
}
state->work_JTJ = gsl_matrix_alloc(p, p);
if (state->work_JTJ == NULL)
{
GSL_ERROR_NULL ("failed to allocate space for JTJ workspace",
GSL_ENOMEM);
}
state->rhs = gsl_vector_alloc(p);
if (state->rhs == NULL)
{
GSL_ERROR_NULL ("failed to allocate space for rhs", GSL_ENOMEM);
}
state->perm = gsl_permutation_alloc(p);
if (state->perm == NULL)
{
GSL_ERROR_NULL ("failed to allocate space for perm", GSL_ENOMEM);
}
state->work3p = gsl_vector_alloc(3 * p);
if (state->work3p == NULL)
{
GSL_ERROR_NULL ("failed to allocate space for work3p", GSL_ENOMEM);
}
state->workn = gsl_vector_alloc(n);
if (state->workn == NULL)
{
GSL_ERROR_NULL ("failed to allocate space for workn", GSL_ENOMEM);
}
state->mu = -1.0;
return state;
}
static void
cholesky_free(void *vstate)
{
cholesky_state_t *state = (cholesky_state_t *) vstate;
if (state->JTJ)
gsl_matrix_free(state->JTJ);
if (state->work_JTJ)
gsl_matrix_free(state->work_JTJ);
if (state->rhs)
gsl_vector_free(state->rhs);
if (state->perm)
gsl_permutation_free(state->perm);
if (state->work3p)
gsl_vector_free(state->work3p);
if (state->workn)
gsl_vector_free(state->workn);
free(state);
}
static int
cholesky_init(const void * vtrust_state, void * vstate)
{
const gsl_multilarge_nlinear_trust_state *trust_state =
(const gsl_multilarge_nlinear_trust_state *) vtrust_state;
cholesky_state_t *state = (cholesky_state_t *) vstate;
/* store J^T J normal equations matrix */
gsl_matrix_tricpy('L', 1, state->JTJ, trust_state->JTJ);
return GSL_SUCCESS;
}
/*
cholesky_presolve()
Compute the modified Cholesky decomposition of J^T J + mu D^T D.
Modified Cholesky is used in case mu = 0 and there are rounding
errors in forming J^T J which could lead to an indefinite matrix.
Inputs: mu - LM parameter
vstate - workspace
Notes:
1) On output, state->work_JTJ contains the Cholesky decomposition of
J^T J + mu D^T D
*/
static int
cholesky_presolve(const double mu, const void * vtrust_state, void * vstate)
{
const gsl_multilarge_nlinear_trust_state *trust_state =
(const gsl_multilarge_nlinear_trust_state *) vtrust_state;
cholesky_state_t *state = (cholesky_state_t *) vstate;
gsl_matrix *JTJ = state->work_JTJ;
const gsl_vector *diag = trust_state->diag;
int status;
/* copy lower triangle of A to workspace */
gsl_matrix_tricpy('L', 1, JTJ, state->JTJ);
/* augment normal equations: A -> A + mu D^T D */
status = cholesky_regularize(mu, diag, JTJ, state);
if (status)
return status;
/* compute modified Cholesky decomposition */
status = gsl_linalg_mcholesky_decomp(JTJ, state->perm, NULL);
if (status)
return status;
state->mu = mu;
return GSL_SUCCESS;
}
/*
cholesky_solve()
Compute (J^T J + mu D^T D) x = -g
where g = J^T f
Inputs: g - right hand side vector g, size p
x - (output) solution vector
vstate - cholesky workspace
*/
static int
cholesky_solve(const gsl_vector * g, gsl_vector *x,
const void * vtrust_state, void *vstate)
{
cholesky_state_t *state = (cholesky_state_t *) vstate;
int status;
status = cholesky_solve_rhs(g, x, state);
if (status)
return status;
/* reverse direction to go downhill */
gsl_vector_scale(x, -1.0);
(void) vtrust_state;
return GSL_SUCCESS;
}
static int
cholesky_rcond(double * rcond, const gsl_matrix * JTJ, void * vstate)
{
int status;
cholesky_state_t *state = (cholesky_state_t *) vstate;
double rcond_JTJ;
/* its possible the current Cholesky decomposition is from the previous
* iteration so do a new one to be sure we use the right Jacobian */
/* copy lower triangle of JTJ to workspace */
gsl_matrix_tricpy('L', 1, state->work_JTJ, JTJ);
/* compute modified Cholesky decomposition */
status = gsl_linalg_mcholesky_decomp(state->work_JTJ, state->perm, NULL);
if (status)
return status;
status = gsl_linalg_mcholesky_rcond(state->work_JTJ, state->perm, &rcond_JTJ, state->work3p);
if (status == GSL_SUCCESS)
*rcond = sqrt(rcond_JTJ);
return status;
}
static int
cholesky_covar(const gsl_matrix * JTJ, gsl_matrix * covar, void * vstate)
{
int status;
cholesky_state_t *state = (cholesky_state_t *) vstate;
/* its possible the current Cholesky decomposition is from the previous
* iteration so do a new one to be sure we use the right Jacobian */
/* copy lower triangle of JTJ to workspace */
gsl_matrix_tricpy('L', 1, state->work_JTJ, JTJ);
/* compute modified Cholesky decomposition */
status = gsl_linalg_mcholesky_decomp(state->work_JTJ, state->perm, NULL);
if (status)
return status;
status = gsl_linalg_mcholesky_invert(state->work_JTJ, state->perm, covar);
if (status)
return status;
return GSL_SUCCESS;
}
/* solve: (J^T J + mu D^T D) x = b */
static int
cholesky_solve_rhs(const gsl_vector * b, gsl_vector *x, cholesky_state_t *state)
{
int status;
gsl_matrix *JTJ = state->work_JTJ;
status = gsl_linalg_mcholesky_solve(JTJ, state->perm, b, x);
if (status)
return status;
return GSL_SUCCESS;
}
/* A <- A + mu D^T D */
static int
cholesky_regularize(const double mu, const gsl_vector * diag, gsl_matrix * A,
cholesky_state_t * state)
{
(void) state;
if (mu != 0.0)
{
size_t i;
for (i = 0; i < diag->size; ++i)
{
double di = gsl_vector_get(diag, i);
double *Aii = gsl_matrix_ptr(A, i, i);
*Aii += mu * di * di;
}
}
return GSL_SUCCESS;
}
static const gsl_multilarge_nlinear_solver cholesky_type =
{
"cholesky",
cholesky_alloc,
cholesky_init,
cholesky_presolve,
cholesky_solve,
cholesky_rcond,
cholesky_covar,
cholesky_free
};
const gsl_multilarge_nlinear_solver *gsl_multilarge_nlinear_solver_cholesky = &cholesky_type;