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/* linalg/sytd.c
*
* Copyright (C) 2001, 2007 Brian Gough
* Copyright (C) 2019 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.
*/
/* Factorise a symmetric matrix A into
*
* A = Q T Q'
*
* where Q is orthogonal and T is symmetric tridiagonal. Only the
* diagonal and lower triangular part of A is referenced and modified.
*
* On exit, T is stored in the diagonal and first subdiagonal of
* A. Since T is symmetric the upper diagonal is not stored.
*
* Q is stored as a packed set of Householder transformations in the
* lower triangular part of the input matrix below the first subdiagonal.
*
* The full matrix for Q can be obtained as the product
*
* Q = Q_1 Q_2 ... Q_(N-2)
*
* where
*
* Q_i = (I - tau_i * v_i * v_i')
*
* and where v_i is a Householder vector
*
* v_i = [0, ... , 0, 1, A(i+1,i), A(i+2,i), ... , A(N,i)]
*
* This storage scheme is the same as in LAPACK. See LAPACK's
* ssytd2.f for details.
*
* See Golub & Van Loan, "Matrix Computations" (3rd ed), Section 8.3
*
* Note: this description uses 1-based indices. The code below uses
* 0-based indices
*/
#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>
int
gsl_linalg_symmtd_decomp (gsl_matrix * A, gsl_vector * tau)
{
if (A->size1 != A->size2)
{
GSL_ERROR ("symmetric tridiagonal decomposition requires square matrix",
GSL_ENOTSQR);
}
else if (tau->size + 1 != A->size1)
{
GSL_ERROR ("size of tau must be N-1", GSL_EBADLEN);
}
else
{
const size_t N = A->size1;
size_t i;
for (i = 0 ; i < N - 2; i++)
{
gsl_vector_view v = gsl_matrix_subcolumn (A, i, i + 1, N - i - 1);
double tau_i = gsl_linalg_householder_transform (&v.vector);
/* Apply the transformation H^T A H to the remaining columns */
if (tau_i != 0.0)
{
gsl_matrix_view m = gsl_matrix_submatrix (A, i + 1, i + 1, N - i - 1, N - i - 1);
double ei = gsl_vector_get(&v.vector, 0);
gsl_vector_view x = gsl_vector_subvector (tau, i, N - i - 1);
gsl_vector_set (&v.vector, 0, 1.0);
/* x = tau * A * v */
gsl_blas_dsymv (CblasLower, tau_i, &m.matrix, &v.vector, 0.0, &x.vector);
/* w = x - (1/2) tau * (x' * v) * v */
{
double xv, alpha;
gsl_blas_ddot(&x.vector, &v.vector, &xv);
alpha = -0.5 * tau_i * xv;
gsl_blas_daxpy(alpha, &v.vector, &x.vector);
}
/* apply the transformation A = A - v w' - w v' */
gsl_blas_dsyr2(CblasLower, -1.0, &v.vector, &x.vector, &m.matrix);
gsl_vector_set (&v.vector, 0, ei);
}
gsl_vector_set (tau, i, tau_i);
}
return GSL_SUCCESS;
}
}
/* Form the orthogonal matrix Q from the packed QR matrix */
int
gsl_linalg_symmtd_unpack (const gsl_matrix * A,
const gsl_vector * tau,
gsl_matrix * Q,
gsl_vector * diag,
gsl_vector * sdiag)
{
if (A->size1 != A->size2)
{
GSL_ERROR ("matrix A must be square", GSL_ENOTSQR);
}
else if (tau->size + 1 != A->size1)
{
GSL_ERROR ("size of tau must be (matrix size - 1)", GSL_EBADLEN);
}
else if (Q->size1 != A->size1 || Q->size2 != A->size1)
{
GSL_ERROR ("size of Q must match size of A", GSL_EBADLEN);
}
else if (diag->size != A->size1)
{
GSL_ERROR ("size of diagonal must match size of A", GSL_EBADLEN);
}
else if (sdiag->size + 1 != A->size1)
{
GSL_ERROR ("size of subdiagonal must be (matrix size - 1)", GSL_EBADLEN);
}
else
{
const size_t N = A->size1;
gsl_vector_const_view d = gsl_matrix_const_diagonal(A);;
gsl_vector_const_view sd = gsl_matrix_const_subdiagonal(A, 1);;
size_t i;
/* Initialize Q to the identity */
gsl_matrix_set_identity (Q);
for (i = N - 2; i-- > 0;)
{
gsl_vector_const_view h = gsl_matrix_const_subcolumn (A, i, i + 1, N - i - 1);
double ti = gsl_vector_get (tau, i);
gsl_matrix_view m = gsl_matrix_submatrix (Q, i + 1, i + 1, N - i - 1, N - i - 1);
gsl_vector_view work = gsl_vector_subvector(diag, 0, N - i - 1);
double * ptr = gsl_vector_ptr((gsl_vector *) &h.vector, 0);
double tmp = *ptr;
*ptr = 1.0;
gsl_linalg_householder_left (ti, &h.vector, &m.matrix, &work.vector);
*ptr = tmp;
}
/* copy diagonal into diag */
gsl_vector_memcpy(diag, &d.vector);
/* copy subdiagonal into sd */
gsl_vector_memcpy(sdiag, &sd.vector);
return GSL_SUCCESS;
}
}
int
gsl_linalg_symmtd_unpack_T (const gsl_matrix * A,
gsl_vector * diag,
gsl_vector * sdiag)
{
if (A->size1 != A->size2)
{
GSL_ERROR ("matrix A must be square", GSL_ENOTSQR);
}
else if (diag->size != A->size1)
{
GSL_ERROR ("size of diagonal must match size of A", GSL_EBADLEN);
}
else if (sdiag->size + 1 != A->size1)
{
GSL_ERROR ("size of subdiagonal must be (matrix size - 1)", GSL_EBADLEN);
}
else
{
gsl_vector_const_view d = gsl_matrix_const_diagonal(A);;
gsl_vector_const_view sd = gsl_matrix_const_subdiagonal(A, 1);;
/* copy diagonal into diag */
gsl_vector_memcpy(diag, &d.vector);
/* copy subdiagonal into sd */
gsl_vector_memcpy(sdiag, &sd.vector);
return GSL_SUCCESS;
}
}
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