/* linalg/hermtd.c
*
* Copyright (C) 2001, 2007, 2009 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.
*/
/* Factorise a hermitian matrix A into
*
* A = U T U'
*
* where U is unitary and T is real 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.
*
* U 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 U can be obtained as the product
*
* U = U_N ... U_2 U_1
*
* where
*
* U_i = (I - tau_i * v_i * v_i')
*
* and where v_i is a Householder vector
*
* v_i = [0, ..., 0, 1, A(i+2,i), A(i+3,i), ... , A(N,i)]
*
* This storage scheme is the same as in LAPACK. See LAPACK's
* chetd2.f for details.
*
* See Golub & Van Loan, "Matrix Computations" (3rd ed), Section 8.3 */
#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_complex_math.h>
#include <gsl/gsl_linalg.h>
int
gsl_linalg_hermtd_decomp (gsl_matrix_complex * A, gsl_vector_complex * tau)
{
if (A->size1 != A->size2)
{
GSL_ERROR ("hermitian tridiagonal decomposition requires square matrix",
GSL_ENOTSQR);
}
else if (tau->size + 1 != A->size1)
{
GSL_ERROR ("size of tau must be (matrix size - 1)", GSL_EBADLEN);
}
else
{
const size_t N = A->size1;
size_t i;
const gsl_complex zero = gsl_complex_rect (0.0, 0.0);
const gsl_complex one = gsl_complex_rect (1.0, 0.0);
const gsl_complex neg_one = gsl_complex_rect (-1.0, 0.0);
for (i = 0 ; i < N - 1; i++)
{
gsl_vector_complex_view c = gsl_matrix_complex_column (A, i);
gsl_vector_complex_view v = gsl_vector_complex_subvector (&c.vector, i + 1, N - (i + 1));
gsl_complex tau_i = gsl_linalg_complex_householder_transform (&v.vector);
/* Apply the transformation H^T A H to the remaining columns */
if ((i + 1) < (N - 1)
&& !(GSL_REAL(tau_i) == 0.0 && GSL_IMAG(tau_i) == 0.0))
{
gsl_matrix_complex_view m =
gsl_matrix_complex_submatrix (A, i + 1, i + 1,
N - (i+1), N - (i+1));
gsl_complex ei = gsl_vector_complex_get(&v.vector, 0);
gsl_vector_complex_view x = gsl_vector_complex_subvector (tau, i, N-(i+1));
gsl_vector_complex_set (&v.vector, 0, one);
/* x = tau * A * v */
gsl_blas_zhemv (CblasLower, tau_i, &m.matrix, &v.vector, zero, &x.vector);
/* w = x - (1/2) tau * (x' * v) * v */
{
gsl_complex xv, txv, alpha;
gsl_blas_zdotc(&x.vector, &v.vector, &xv);
txv = gsl_complex_mul(tau_i, xv);
alpha = gsl_complex_mul_real(txv, -0.5);
gsl_blas_zaxpy(alpha, &v.vector, &x.vector);
}
/* apply the transformation A = A - v w' - w v' */
gsl_blas_zher2(CblasLower, neg_one, &v.vector, &x.vector, &m.matrix);
gsl_vector_complex_set (&v.vector, 0, ei);
}
gsl_vector_complex_set (tau, i, tau_i);
}
return GSL_SUCCESS;
}
}
/* Form the orthogonal matrix U from the packed QR matrix */
int
gsl_linalg_hermtd_unpack (const gsl_matrix_complex * A,
const gsl_vector_complex * tau,
gsl_matrix_complex * U,
gsl_vector * diag,
gsl_vector * sdiag)
{
if (A->size1 != A->size2)
{
GSL_ERROR ("matrix A must be sqaure", GSL_ENOTSQR);
}
else if (tau->size + 1 != A->size1)
{
GSL_ERROR ("size of tau must be (matrix size - 1)", GSL_EBADLEN);
}
else if (U->size1 != A->size1 || U->size2 != A->size1)
{
GSL_ERROR ("size of U 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;
size_t i;
/* Initialize U to the identity */
gsl_matrix_complex_set_identity (U);
for (i = N - 1; i-- > 0;)
{
gsl_complex ti = gsl_vector_complex_get (tau, i);
gsl_vector_complex_const_view c = gsl_matrix_complex_const_column (A, i);
gsl_vector_complex_const_view h =
gsl_vector_complex_const_subvector (&c.vector, i + 1, N - (i+1));
gsl_matrix_complex_view m =
gsl_matrix_complex_submatrix (U, i + 1, i + 1, N-(i+1), N-(i+1));
gsl_linalg_complex_householder_hm (ti, &h.vector, &m.matrix);
}
/* Copy diagonal into diag */
for (i = 0; i < N; i++)
{
gsl_complex Aii = gsl_matrix_complex_get (A, i, i);
gsl_vector_set (diag, i, GSL_REAL(Aii));
}
/* Copy subdiagonal into sdiag */
for (i = 0; i < N - 1; i++)
{
gsl_complex Aji = gsl_matrix_complex_get (A, i+1, i);
gsl_vector_set (sdiag, i, GSL_REAL(Aji));
}
return GSL_SUCCESS;
}
}
int
gsl_linalg_hermtd_unpack_T (const gsl_matrix_complex * A,
gsl_vector * diag,
gsl_vector * sdiag)
{
if (A->size1 != A->size2)
{
GSL_ERROR ("matrix A must be sqaure", 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
{
const size_t N = A->size1;
size_t i;
/* Copy diagonal into diag */
for (i = 0; i < N; i++)
{
gsl_complex Aii = gsl_matrix_complex_get (A, i, i);
gsl_vector_set (diag, i, GSL_REAL(Aii));
}
/* Copy subdiagonal into sd */
for (i = 0; i < N - 1; i++)
{
gsl_complex Aji = gsl_matrix_complex_get (A, i+1, i);
gsl_vector_set (sdiag, i, GSL_REAL(Aji));
}
return GSL_SUCCESS;
}
}