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/* linalg/hh.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman, 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.
*/
/* Author: G. Jungman */
#include <config.h>
#include <stdlib.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_linalg.h>
#define REAL double
/* [Engeln-Mullges + Uhlig, Alg. 4.42]
*/
int
gsl_linalg_HH_solve (gsl_matrix * A, const gsl_vector * b, gsl_vector * x)
{
if (A->size1 > A->size2)
{
/* System is underdetermined. */
GSL_ERROR ("System is underdetermined", GSL_EINVAL);
}
else if (A->size2 != x->size)
{
GSL_ERROR ("matrix and vector sizes must be equal", GSL_EBADLEN);
}
else
{
int status ;
gsl_vector_memcpy (x, b);
status = gsl_linalg_HH_svx (A, x);
return status ;
}
}
int
gsl_linalg_HH_svx (gsl_matrix * A, gsl_vector * x)
{
if (A->size1 > A->size2)
{
/* System is underdetermined. */
GSL_ERROR ("System is underdetermined", GSL_EINVAL);
}
else if (A->size2 != x->size)
{
GSL_ERROR ("matrix and vector sizes must be equal", GSL_EBADLEN);
}
else
{
const size_t N = A->size1;
const size_t M = A->size2;
size_t i, j, k;
REAL *d = (REAL *) malloc (N * sizeof (REAL));
if (d == 0)
{
GSL_ERROR ("could not allocate memory for workspace", GSL_ENOMEM);
}
/* Perform Householder transformation. */
for (i = 0; i < N; i++)
{
const REAL aii = gsl_matrix_get (A, i, i);
REAL alpha;
REAL f;
REAL ak;
REAL max_norm = 0.0;
REAL r = 0.0;
for (k = i; k < M; k++)
{
REAL aki = gsl_matrix_get (A, k, i);
r += aki * aki;
}
if (r == 0.0)
{
/* Rank of matrix is less than size1. */
free (d);
GSL_ERROR ("matrix is rank deficient", GSL_ESING);
}
alpha = sqrt (r) * GSL_SIGN (aii);
ak = 1.0 / (r + alpha * aii);
gsl_matrix_set (A, i, i, aii + alpha);
d[i] = -alpha;
for (k = i + 1; k < N; k++)
{
REAL norm = 0.0;
f = 0.0;
for (j = i; j < M; j++)
{
REAL ajk = gsl_matrix_get (A, j, k);
REAL aji = gsl_matrix_get (A, j, i);
norm += ajk * ajk;
f += ajk * aji;
}
max_norm = GSL_MAX (max_norm, norm);
f *= ak;
for (j = i; j < M; j++)
{
REAL ajk = gsl_matrix_get (A, j, k);
REAL aji = gsl_matrix_get (A, j, i);
gsl_matrix_set (A, j, k, ajk - f * aji);
}
}
if (fabs (alpha) < 2.0 * GSL_DBL_EPSILON * sqrt (max_norm))
{
/* Apparent singularity. */
free (d);
GSL_ERROR("apparent singularity detected", GSL_ESING);
}
/* Perform update of RHS. */
f = 0.0;
for (j = i; j < M; j++)
{
f += gsl_vector_get (x, j) * gsl_matrix_get (A, j, i);
}
f *= ak;
for (j = i; j < M; j++)
{
REAL xj = gsl_vector_get (x, j);
REAL aji = gsl_matrix_get (A, j, i);
gsl_vector_set (x, j, xj - f * aji);
}
}
/* Perform back-substitution. */
for (i = N; i > 0 && i--;)
{
REAL xi = gsl_vector_get (x, i);
REAL sum = 0.0;
for (k = i + 1; k < N; k++)
{
sum += gsl_matrix_get (A, i, k) * gsl_vector_get (x, k);
}
gsl_vector_set (x, i, (xi - sum) / d[i]);
}
free (d);
return GSL_SUCCESS;
}
}
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