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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324
|
/* 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;
|