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/* linalg/ql.c
*
* 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.
*/
#include <config.h>
#include <stdlib.h>
#include <string.h>
#include <gsl/gsl_linalg.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_blas.h>
/* Factorise a general M x N matrix A into
*
* A = Q L
*
* where Q is orthogonal (M x M) and L is lower triangular (M x N).
*
* Q is stored as a packed set of Householder transformations in the
* strict upper triangular part of the input matrix.
*
* L is stored in the diagonal and lower triangle of the input matrix.
*
* The full matrix for Q can be obtained as the product
*
* Q = Q_k .. Q_2 Q_1
*
* where k = MIN(M,N) and
*
* Q_i = (I - tau_i * v_i * v_i')
*
* and where v_i is a Householder vector
*
* v_i = [A(1,N-k+i), A(2,N-k+i), ... , A(M-k+i,N-k+i), 1, 0, ..., 0]
*
* This storage scheme is the same as in LAPACK. */
/*
gsl_linalg_QL_decomp()
Perform QL decomposition of a matrix A
Inputs: A - M-by-N matrix
tau - (output) Householder coefficients, length N
Notes:
1) The K = MIN(M, N) Householder scalars are stored in tau(N-K+1:N)
on output; the rest of tau is used as temporary workspace
*/
int
gsl_linalg_QL_decomp (gsl_matrix * A, gsl_vector * tau)
{
const size_t M = A->size1;
const size_t N = A->size2;
if (tau->size != N)
{
GSL_ERROR ("size of tau must be N", GSL_EBADLEN);
}
else
{
const size_t K = GSL_MIN(M, N);
size_t i;
for (i = 0; i < K; i++)
{
/* compute the Householder transformation to annihilate the (N-K+i)-th
column of the matrix */
gsl_vector_view c = gsl_matrix_subcolumn (A, N - i - 1, 0, M - i);
double * alpha = gsl_matrix_ptr(A, M - i - 1, N - i - 1);
double tau_j = gsl_linalg_householder_transform2 (alpha, &(c.vector));
/* apply the transformation to A(1:M-i,1:N-i-2) from the left */
if (i + 1 < N)
{
gsl_vector_view work = gsl_vector_subvector(tau, 0, N - i - 1);
gsl_matrix_view m = gsl_matrix_submatrix (A, 0, 0, M - i, N - i - 1);
double tmp = *alpha;
*alpha = 1.0;
gsl_linalg_householder_left (tau_j, &(c.vector), &(m.matrix), &work.vector);
*alpha = tmp;
}
gsl_vector_set (tau, N - i - 1, tau_j);
}
return GSL_SUCCESS;
}
}
/* form the orthogonal matrix Q from the packed QL matrix */
int
gsl_linalg_QL_unpack (const gsl_matrix * QL, const gsl_vector * tau, gsl_matrix * Q, gsl_matrix * L)
{
const size_t M = QL->size1;
const size_t N = QL->size2;
if (Q->size1 != M || Q->size2 != M)
{
GSL_ERROR ("Q matrix must be M x M", GSL_ENOTSQR);
}
else if (L->size1 != M || L->size2 != N)
{
GSL_ERROR ("L matrix must be M x N", GSL_ENOTSQR);
}
else if (tau->size != N)
{
GSL_ERROR ("size of tau must be N", GSL_EBADLEN);
}
else
{
const size_t K = GSL_MIN(M, N);
size_t i;
/* initialize Q to the identity */
gsl_matrix_set_identity (Q);
for (i = 0; i < K; ++i)
{
gsl_vector_const_view h = gsl_matrix_const_subcolumn (QL, N - K + i, 0, M - K + i + 1);
gsl_matrix_view m = gsl_matrix_submatrix (Q, 0, 0, M - K + i + 1, M - K + i + 1);
gsl_vector_view work = gsl_matrix_subcolumn(L, 0, 0, M - K + i + 1);
double ti = gsl_vector_get (tau, N - K + i);
double * ptr = gsl_matrix_ptr((gsl_matrix *) QL, M - K + i, N - K + i);
double tmp = *ptr;
*ptr = 1.0;
gsl_linalg_householder_left (ti, &h.vector, &m.matrix, &work.vector);
*ptr = tmp;
}
/* form the left triangular matrix L from a packed QL matrix */
gsl_matrix_set_zero(L);
if (M >= N)
{
gsl_matrix_const_view src = gsl_matrix_const_submatrix(QL, M - N, 0, N, N);
gsl_matrix_view dest = gsl_matrix_submatrix(L, M - N, 0, N, N);
gsl_matrix_tricpy(CblasLower, CblasNonUnit, &dest.matrix, &src.matrix);
}
else
{
gsl_matrix_const_view src1 = gsl_matrix_const_submatrix(QL, 0, 0, M, N - M);
gsl_matrix_view dest1 = gsl_matrix_submatrix(L, 0, 0, M, N - M);
gsl_matrix_const_view src2 = gsl_matrix_const_submatrix(QL, 0, N - M, M, M);
gsl_matrix_view dest2 = gsl_matrix_submatrix(L, 0, N - M, M, M);
gsl_matrix_memcpy(&dest1.matrix, &src1.matrix);
gsl_matrix_tricpy(CblasLower, CblasNonUnit, &dest2.matrix, &src2.matrix);
}
return GSL_SUCCESS;
}
}
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