/* linalg/householder.c
*
* Copyright (C) 1996, 1997, 1998, 1999, 2000, 2004, 2007, 2010 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 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.
*/
#include <config.h>
#include <stdlib.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_linalg.h>
/*
gsl_linalg_householder_transform()
Compute a householder transformation (tau,v) of a vector
x so that P x = [ I - tau*v*v' ] x annihilates x(1:n-1)
Inputs: v - on input, x vector
on output, householder vector v
Notes:
1) on output, v is normalized so that v[0] = 1. The 1 is
not actually stored; instead v[0] = -sign(x[0])*||x|| so
that:
P x = v[0] * e_1
Therefore external routines should take care when applying
the projection matrix P to vectors, taking into account
that v[0] should be 1 when doing so.
*/
double
gsl_linalg_householder_transform (gsl_vector * v)
{
/* replace v[0:n-1] with a householder vector (v[0:n-1]) and
coefficient tau that annihilate v[1:n-1] */
const size_t n = v->size ;
if (n == 1)
{
return 0.0; /* tau = 0 */
}
else
{
double alpha, beta, tau ;
gsl_vector_view x = gsl_vector_subvector (v, 1, n - 1) ;
double xnorm = gsl_blas_dnrm2 (&x.vector);
if (xnorm == 0)
{
return 0.0; /* tau = 0 */
}
alpha = gsl_vector_get (v, 0) ;
beta = - (alpha >= 0.0 ? +1.0 : -1.0) * hypot(alpha, xnorm) ;
tau = (beta - alpha) / beta ;
{
double s = (alpha - beta);
if (fabs(s) > GSL_DBL_MIN)
{
gsl_blas_dscal (1.0 / s, &x.vector);
gsl_vector_set (v, 0, beta) ;
}
else
{
gsl_blas_dscal (GSL_DBL_EPSILON / s, &x.vector);
gsl_blas_dscal (1.0 / GSL_DBL_EPSILON, &x.vector);
gsl_vector_set (v, 0, beta) ;
}
}
return tau;
}
}
int
gsl_linalg_householder_hm (double tau, const gsl_vector * v, gsl_matrix * A)
{
/* applies a householder transformation v,tau to matrix m */
if (tau == 0.0)
{
return GSL_SUCCESS;
}
#ifdef USE_BLAS
{
gsl_vector_const_view v1 = gsl_vector_const_subvector (v, 1, v->size - 1);
gsl_matrix_view A1 = gsl_matrix_submatrix (A, 1, 0, A->size1 - 1, A->size2);
size_t j;
for (j = 0; j < A->size2; j++)
{
double wj = 0.0;
gsl_vector_view A1j = gsl_matrix_column(&A1.matrix, j);
gsl_blas_ddot (&A1j.vector, &v1.vector, &wj);
wj += gsl_matrix_get(A,0,j);
{
double A0j = gsl_matrix_get (A, 0, j);
gsl_matrix_set (A, 0, j, A0j - tau * wj);
}
gsl_blas_daxpy (-tau * wj, &v1.vector, &A1j.vector);
}
}
#else
{
size_t i, j;
for (j = 0; j < A->size2; j++)
{
/* Compute wj = Akj vk */
double wj = gsl_matrix_get(A,0,j);
for (i = 1; i < A->size1; i++) /* note, computed for v(0) = 1 above */
{
wj += gsl_matrix_get(A,i,j) * gsl_vector_get(v,i);
}
/* Aij = Aij - tau vi wj */
/* i = 0 */
{
double A0j = gsl_matrix_get (A, 0, j);
gsl_matrix_set (A, 0, j, A0j - tau * wj);
}
/* i = 1 .. M-1 */
for (i = 1; i < A->size1; i++)
{
double Aij = gsl_matrix_get (A, i, j);
double vi = gsl_vector_get (v, i);
gsl_matrix_set (A, i, j, Aij - tau * vi * wj);
}
}
}
#endif
return GSL_SUCCESS;
}
int
gsl_linalg_householder_mh (double tau, const gsl_vector * v, gsl_matrix * A)
{
/* applies a householder transformation v,tau to matrix m from the
right hand side in order to zero out rows */
if (tau == 0)
return GSL_SUCCESS;
/* A = A - tau w v' */
#ifdef USE_BLAS
{
gsl_vector_const_view v1 = gsl_vector_const_subvector (v, 1, v->size - 1);
gsl_matrix_view A1 = gsl_matrix_submatrix (A, 0, 1, A->size1, A->size2-1);
size_t i;
for (i = 0; i < A->size1; i++)
{
double wi = 0.0;
gsl_vector_view A1i = gsl_matrix_row(&A1.matrix, i);
gsl_blas_ddot (&A1i.vector, &v1.vector, &wi);
wi += gsl_matrix_get(A,i,0);
{
double Ai0 = gsl_matrix_get (A, i, 0);
gsl_matrix_set (A, i, 0, Ai0 - tau * wi);
}
gsl_blas_daxpy(-tau * wi, &v1.vector, &A1i.vector);
}
}
#else
{
size_t i, j;
for (i = 0; i < A->size1; i++)
{
double wi = gsl_matrix_get(A,i,0);
for (j = 1; j < A->size2; j++) /* note, computed for v(0) = 1 above */
{
wi += gsl_matrix_get(A,i,j) * gsl_vector_get(v,j);
}
/* j = 0 */
{
double Ai0 = gsl_matrix_get (A, i, 0);
gsl_matrix_set (A, i, 0, Ai0 - tau * wi);
}
/* j = 1 .. N-1 */
for (j = 1; j < A->size2; j++)
{
double vj = gsl_vector_get (v, j);
double Aij = gsl_matrix_get (A, i, j);
gsl_matrix_set (A, i, j, Aij - tau * wi * vj);
}
}
}
#endif
return GSL_SUCCESS;
}
int
gsl_linalg_householder_hv (double tau, const gsl_vector * v, gsl_vector * w)
{
/* applies a householder transformation v to vector w */
const size_t N = v->size;
if (tau == 0)
return GSL_SUCCESS ;
{
/* compute d = v'w */
double w0 = gsl_vector_get(w,0);
double d1, d;
gsl_vector_const_view v1 = gsl_vector_const_subvector(v, 1, N-1);
gsl_vector_view w1 = gsl_vector_subvector(w, 1, N-1);
/* compute d1 = v(2:n)'w(2:n) */
gsl_blas_ddot (&v1.vector, &w1.vector, &d1);
/* compute d = v'w = w(1) + d1 since v(1) = 1 */
d = w0 + d1;
/* compute w = w - tau (v) (v'w) */
gsl_vector_set (w, 0, w0 - tau * d);
gsl_blas_daxpy (-tau * d, &v1.vector, &w1.vector);
}
return GSL_SUCCESS;
}
int
gsl_linalg_householder_hm1 (double tau, gsl_matrix * A)
{
/* applies a householder transformation v,tau to a matrix being
build up from the identity matrix, using the first column of A as
a householder vector */
if (tau == 0)
{
size_t i,j;
gsl_matrix_set (A, 0, 0, 1.0);
for (j = 1; j < A->size2; j++)
{
gsl_matrix_set (A, 0, j, 0.0);
}
for (i = 1; i < A->size1; i++)
{
gsl_matrix_set (A, i, 0, 0.0);
}
return GSL_SUCCESS;
}
/* w = A' v */
#ifdef USE_BLAS
{
gsl_matrix_view A1 = gsl_matrix_submatrix (A, 1, 0, A->size1 - 1, A->size2);
gsl_vector_view v1 = gsl_matrix_column (&A1.matrix, 0);
size_t j;
for (j = 1; j < A->size2; j++)
{
double wj = 0.0; /* A0j * v0 */
gsl_vector_view A1j = gsl_matrix_column(&A1.matrix, j);
gsl_blas_ddot (&A1j.vector, &v1.vector, &wj);
/* A = A - tau v w' */
gsl_matrix_set (A, 0, j, - tau * wj);
gsl_blas_daxpy(-tau*wj, &v1.vector, &A1j.vector);
}
gsl_blas_dscal(-tau, &v1.vector);
gsl_matrix_set (A, 0, 0, 1.0 - tau);
}
#else
{
size_t i, j;
for (j = 1; j < A->size2; j++)
{
double wj = 0.0; /* A0j * v0 */
for (i = 1; i < A->size1; i++)
{
double vi = gsl_matrix_get(A, i, 0);
wj += gsl_matrix_get(A,i,j) * vi;
}
/* A = A - tau v w' */
gsl_matrix_set (A, 0, j, - tau * wj);
for (i = 1; i < A->size1; i++)
{
double vi = gsl_matrix_get (A, i, 0);
double Aij = gsl_matrix_get (A, i, j);
gsl_matrix_set (A, i, j, Aij - tau * vi * wj);
}
}
for (i = 1; i < A->size1; i++)
{
double vi = gsl_matrix_get(A, i, 0);
gsl_matrix_set(A, i, 0, -tau * vi);
}
gsl_matrix_set (A, 0, 0, 1.0 - tau);
}
#endif
return GSL_SUCCESS;
}