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+/* linalg/lq.c
+ *
+ * Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman, Brian Gough
+ * Copyright (C) 2004 Joerg Wensch, modifications for LQ.
+ *
+ * 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.
+ */
+
+#include <config.h>
+#include <stdlib.h>
+#include <string.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>
+
+#define REAL double
+
+#include "givens.c"
+#include "apply_givens.c"
+
+/* Note: The standard in numerical linear algebra is to solve A x = b
+ * resp. ||A x - b||_2 -> min by QR-decompositions where x, b are
+ * column vectors.
+ *
+ * When the matrix A has a large number of rows it is much more
+ * efficient to work with the transposed matrix A^T and to solve the
+ * system x^T A = b^T resp. ||x^T A - b^T||_2 -> min. This is caused
+ * by the row-oriented format in which GSL stores matrices. Therefore
+ * the QR-decomposition of A has to be replaced by a LQ decomposition
+ * of A^T
+ *
+ * The purpose of this package is to provide the algorithms to compute
+ * the LQ-decomposition and to solve the linear equations resp. least
+ * squares problems. The dimensions N, M of the matrix are switched
+ * because here A will probably be a transposed matrix. We write x^T,
+ * b^T,... for vectors the comments to emphasize that they are row
+ * vectors.
+ *
+ * It may even be useful to transpose your matrix explicitly (assumed
+ * that there are no memory restrictions) because this takes O(M x N)
+ * computing time where the decompostion takes O(M x N^2) computing
+ * time. */
+
+/* Factorise a general N x M matrix A into
+ *
+ * A = L Q
+ *
+ * where Q is orthogonal (M x M) and L is lower triangular (N x M).
+ *
+ * Q is stored as a packed set of Householder transformations in the
+ * strict upper triangular part of the input matrix.
+ *
+ * R 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 = [1, m(i+1,i), m(i+2,i), ... , m(M,i)]
+ *
+ * This storage scheme is the same as in LAPACK. */
+
+int
+gsl_linalg_LQ_decomp (gsl_matrix * A, gsl_vector * tau)
+{
+ const size_t N = A->size1;
+ const size_t M = A->size2;
+
+ if (tau->size != GSL_MIN (M, N))
+ {
+ GSL_ERROR ("size of tau must be MIN(M,N)", GSL_EBADLEN);
+ }
+ else
+ {
+ size_t i;
+
+ for (i = 0; i < GSL_MIN (M, N); i++)
+ {
+ /* Compute the Householder transformation to reduce the j-th
+ column of the matrix to a multiple of the j-th unit vector */
+
+ gsl_vector_view c_full = gsl_matrix_row (A, i);
+ gsl_vector_view c = gsl_vector_subvector (&(c_full.vector), i, M-i);
+
+ double tau_i = gsl_linalg_householder_transform (&(c.vector));
+
+ gsl_vector_set (tau, i, tau_i);
+
+ /* Apply the transformation to the remaining columns and
+ update the norms */
+
+ if (i + 1 < N)
+ {
+ gsl_matrix_view m = gsl_matrix_submatrix (A, i + 1, i, N - (i + 1), M - i );
+ gsl_linalg_householder_mh (tau_i, &(c.vector), &(m.matrix));
+ }
+ }
+
+ return GSL_SUCCESS;
+ }
+}
+
+/* Solves the system x^T A = b^T using the LQ factorisation,
+
+ * x^T L = b^T Q^T
+ *
+ * to obtain x. Based on SLATEC code.
+ */
+
+
+int
+gsl_linalg_LQ_solve_T (const gsl_matrix * LQ, const gsl_vector * tau, const gsl_vector * b, gsl_vector * x)
+{
+ if (LQ->size1 != LQ->size2)
+ {
+ GSL_ERROR ("LQ matrix must be square", GSL_ENOTSQR);
+ }
+ else if (LQ->size2 != b->size)
+ {
+ GSL_ERROR ("matrix size must match b size", GSL_EBADLEN);
+ }
+ else if (LQ->size1 != x->size)
+ {
+ GSL_ERROR ("matrix size must match solution size", GSL_EBADLEN);
+ }
+ else
+ {
+ /* Copy x <- b */
+
+ gsl_vector_memcpy (x, b);
+
+ /* Solve for x */
+
+ gsl_linalg_LQ_svx_T (LQ, tau, x);
+
+ return GSL_SUCCESS;
+ }
+}
+
+/* Solves the system x^T A = b^T in place using the LQ factorisation,
+ *
+ * x^T L = b^T Q^T
+ *
+ * to obtain x. Based on SLATEC code.
+ */
+
+int
+gsl_linalg_LQ_svx_T (const gsl_matrix * LQ, const gsl_vector * tau, gsl_vector * x)
+{
+
+ if (LQ->size1 != LQ->size2)
+ {
+ GSL_ERROR ("LQ matrix must be square", GSL_ENOTSQR);
+ }
+ else if (LQ->size1 != x->size)
+ {
+ GSL_ERROR ("matrix size must match x/rhs size", GSL_EBADLEN);
+ }
+ else
+ {
+ /* compute rhs = Q^T b */
+
+ gsl_linalg_LQ_vecQT (LQ, tau, x);
+
+ /* Solve R x = rhs, storing x in-place */
+
+ gsl_blas_dtrsv (CblasLower, CblasTrans, CblasNonUnit, LQ, x);
+
+ return GSL_SUCCESS;
+ }
+}
+
+
+/* Find the least squares solution to the overdetermined system
+ *
+ * x^T A = b^T
+ *
+ * for M >= N using the LQ factorization A = L Q.
+ */
+
+int
+gsl_linalg_LQ_lssolve_T (const gsl_matrix * LQ, const gsl_vector * tau, const gsl_vector * b, gsl_vector * x, gsl_vector * residual)
+{
+ const size_t N = LQ->size1;
+ const size_t M = LQ->size2;
+
+ if (M < N)
+ {
+ GSL_ERROR ("LQ matrix must have M>=N", GSL_EBADLEN);
+ }
+ else if (M != b->size)
+ {
+ GSL_ERROR ("matrix size must match b size", GSL_EBADLEN);
+ }
+ else if (N != x->size)
+ {
+ GSL_ERROR ("matrix size must match solution size", GSL_EBADLEN);
+ }
+ else if (M != residual->size)
+ {
+ GSL_ERROR ("matrix size must match residual size", GSL_EBADLEN);
+ }
+ else
+ {
+ gsl_matrix_const_view L = gsl_matrix_const_submatrix (LQ, 0, 0, N, N);
+ gsl_vector_view c = gsl_vector_subvector(residual, 0, N);
+
+ gsl_vector_memcpy(residual, b);
+
+ /* compute rhs = b^T Q^T */
+
+ gsl_linalg_LQ_vecQT (LQ, tau, residual);
+
+ /* Solve x^T L = rhs */
+
+ gsl_vector_memcpy(x, &(c.vector));
+
+ gsl_blas_dtrsv (CblasLower, CblasTrans, CblasNonUnit, &(L.matrix), x);
+
+ /* Compute residual = b^T - x^T A = (b^T Q^T - x^T L) Q */
+
+ gsl_vector_set_zero(&(c.vector));
+
+ gsl_linalg_LQ_vecQ(LQ, tau, residual);
+
+ return GSL_SUCCESS;
+ }
+}
+
+
+int
+gsl_linalg_LQ_Lsolve_T (const gsl_matrix * LQ, const gsl_vector * b, gsl_vector * x)
+{
+ if (LQ->size1 != LQ->size2)
+ {
+ GSL_ERROR ("LQ matrix must be square", GSL_ENOTSQR);
+ }
+ else if (LQ->size1 != b->size)
+ {
+ GSL_ERROR ("matrix size must match b size", GSL_EBADLEN);
+ }
+ else if (LQ->size1 != x->size)
+ {
+ GSL_ERROR ("matrix size must match x size", GSL_EBADLEN);
+ }
+ else
+ {
+ /* Copy x <- b */
+
+ gsl_vector_memcpy (x, b);
+
+ /* Solve R x = b, storing x in-place */
+
+ gsl_blas_dtrsv (CblasLower, CblasTrans, CblasNonUnit, LQ, x);
+
+ return GSL_SUCCESS;
+ }
+}
+
+
+int
+gsl_linalg_LQ_Lsvx_T (const gsl_matrix * LQ, gsl_vector * x)
+{
+ if (LQ->size1 != LQ->size2)
+ {
+ GSL_ERROR ("LQ matrix must be square", GSL_ENOTSQR);
+ }
+ else if (LQ->size2 != x->size)
+ {
+ GSL_ERROR ("matrix size must match rhs size", GSL_EBADLEN);
+ }
+ else
+ {
+ /* Solve x^T L = b^T, storing x in-place */
+
+ gsl_blas_dtrsv (CblasLower, CblasTrans, CblasNonUnit, LQ, x);
+
+ return GSL_SUCCESS;
+ }
+}
+
+int
+gsl_linalg_L_solve_T (const gsl_matrix * L, const gsl_vector * b, gsl_vector * x)
+{
+ if (L->size1 != L->size2)
+ {
+ GSL_ERROR ("R matrix must be square", GSL_ENOTSQR);
+ }
+ else if (L->size2 != b->size)
+ {
+ GSL_ERROR ("matrix size must match b size", GSL_EBADLEN);
+ }
+ else if (L->size1 != x->size)
+ {
+ GSL_ERROR ("matrix size must match solution size", GSL_EBADLEN);
+ }
+ else
+ {
+ /* Copy x <- b */
+
+ gsl_vector_memcpy (x, b);
+
+ /* Solve R x = b, storing x inplace in b */
+
+ gsl_blas_dtrsv (CblasLower, CblasTrans, CblasNonUnit, L, x);
+
+ return GSL_SUCCESS;
+ }
+}
+
+
+
+
+int
+gsl_linalg_LQ_vecQT (const gsl_matrix * LQ, const gsl_vector * tau, gsl_vector * v)
+{
+ const size_t N = LQ->size1;
+ const size_t M = LQ->size2;
+
+ if (tau->size != GSL_MIN (M, N))
+ {
+ GSL_ERROR ("size of tau must be MIN(M,N)", GSL_EBADLEN);
+ }
+ else if (v->size != M)
+ {
+ GSL_ERROR ("vector size must be M", GSL_EBADLEN);
+ }
+ else
+ {
+ size_t i;
+
+ /* compute v Q^T */
+
+ for (i = 0; i < GSL_MIN (M, N); i++)
+ {
+ gsl_vector_const_view c = gsl_matrix_const_row (LQ, i);
+ gsl_vector_const_view h = gsl_vector_const_subvector (&(c.vector),
+ i, M - i);
+ gsl_vector_view w = gsl_vector_subvector (v, i, M - i);
+ double ti = gsl_vector_get (tau, i);
+ gsl_linalg_householder_hv (ti, &(h.vector), &(w.vector));
+ }
+ return GSL_SUCCESS;
+ }
+}
+
+int
+gsl_linalg_LQ_vecQ (const gsl_matrix * LQ, const gsl_vector * tau, gsl_vector * v)
+{
+ const size_t N = LQ->size1;
+ const size_t M = LQ->size2;
+
+ if (tau->size != GSL_MIN (M, N))
+ {
+ GSL_ERROR ("size of tau must be MIN(M,N)", GSL_EBADLEN);
+ }
+ else if (v->size != M)
+ {
+ GSL_ERROR ("vector size must be M", GSL_EBADLEN);
+ }
+ else
+ {
+ size_t i;
+
+ /* compute v Q^T */
+
+ for (i = GSL_MIN (M, N); i > 0 && i--;)
+ {
+ gsl_vector_const_view c = gsl_matrix_const_row (LQ, i);
+ gsl_vector_const_view h = gsl_vector_const_subvector (&(c.vector),
+ i, M - i);
+ gsl_vector_view w = gsl_vector_subvector (v, i, M - i);
+ double ti = gsl_vector_get (tau, i);
+ gsl_linalg_householder_hv (ti, &(h.vector), &(w.vector));
+ }
+ return GSL_SUCCESS;
+ }
+}
+
+
+/* Form the orthogonal matrix Q from the packed LQ matrix */
+
+int
+gsl_linalg_LQ_unpack (const gsl_matrix * LQ, const gsl_vector * tau, gsl_matrix * Q, gsl_matrix * L)
+{
+ const size_t N = LQ->size1;
+ const size_t M = LQ->size2;
+
+ if (Q->size1 != M || Q->size2 != M)
+ {
+ GSL_ERROR ("Q matrix must be M x M", GSL_ENOTSQR);
+ }
+ else if (L->size1 != N || L->size2 != M)
+ {
+ GSL_ERROR ("R matrix must be N x M", GSL_ENOTSQR);
+ }
+ else if (tau->size != GSL_MIN (M, N))
+ {
+ GSL_ERROR ("size of tau must be MIN(M,N)", GSL_EBADLEN);
+ }
+ else
+ {
+ size_t i, j, l_border;
+
+ /* Initialize Q to the identity */
+
+ gsl_matrix_set_identity (Q);
+
+ for (i = GSL_MIN (M, N); i > 0 && i--;)
+ {
+ gsl_vector_const_view c = gsl_matrix_const_row (LQ, i);
+ gsl_vector_const_view h = gsl_vector_const_subvector (&c.vector,
+ i, M - i);
+ gsl_matrix_view m = gsl_matrix_submatrix (Q, i, i, M - i, M - i);
+ double ti = gsl_vector_get (tau, i);
+ gsl_linalg_householder_mh (ti, &h.vector, &m.matrix);
+ }
+
+ /* Form the lower triangular matrix L from a packed LQ matrix */
+
+ for (i = 0; i < N; i++)
+ {
+ l_border=GSL_MIN(i,M-1);
+ for (j = 0; j <= l_border ; j++)
+ gsl_matrix_set (L, i, j, gsl_matrix_get (LQ, i, j));
+
+ for (j = l_border+1; j < M; j++)
+ gsl_matrix_set (L, i, j, 0.0);
+ }
+
+ return GSL_SUCCESS;
+ }
+}
+
+
+/* Update a LQ factorisation for A= L Q , A' = A + v u^T,
+
+ * L' Q' = LQ + v u^T
+ * = (L + v u^T Q^T) Q
+ * = (L + v w^T) Q
+ *
+ * where w = Q u.
+ *
+ * Algorithm from Golub and Van Loan, "Matrix Computations", Section
+ * 12.5 (Updating Matrix Factorizations, Rank-One Changes)
+ */
+
+int
+gsl_linalg_LQ_update (gsl_matrix * Q, gsl_matrix * L,
+ const gsl_vector * v, gsl_vector * w)
+{
+ const size_t N = L->size1;
+ const size_t M = L->size2;
+
+ if (Q->size1 != M || Q->size2 != M)
+ {
+ GSL_ERROR ("Q matrix must be N x N if L is M x N", GSL_ENOTSQR);
+ }
+ else if (w->size != M)
+ {
+ GSL_ERROR ("w must be length N if L is M x N", GSL_EBADLEN);
+ }
+ else if (v->size != N)
+ {
+ GSL_ERROR ("v must be length M if L is M x N", GSL_EBADLEN);
+ }
+ else
+ {
+ size_t j, k;
+ double w0;
+
+ /* Apply Given's rotations to reduce w to (|w|, 0, 0, ... , 0)
+
+ J_1^T .... J_(n-1)^T w = +/- |w| e_1
+
+ simultaneously applied to L, H = J_1^T ... J^T_(n-1) L
+ so that H is upper Hessenberg. (12.5.2) */
+
+ for (k = M - 1; k > 0; k--)
+ {
+ double c, s;
+ double wk = gsl_vector_get (w, k);
+ double wkm1 = gsl_vector_get (w, k - 1);
+
+ create_givens (wkm1, wk, &c, &s);
+ apply_givens_vec (w, k - 1, k, c, s);
+ apply_givens_lq (M, N, Q, L, k - 1, k, c, s);
+ }
+
+ w0 = gsl_vector_get (w, 0);
+
+ /* Add in v w^T (Equation 12.5.3) */
+
+ for (j = 0; j < N; j++)
+ {
+ double lj0 = gsl_matrix_get (L, j, 0);
+ double vj = gsl_vector_get (v, j);
+ gsl_matrix_set (L, j, 0, lj0 + w0 * vj);
+ }
+
+ /* Apply Givens transformations L' = G_(n-1)^T ... G_1^T H
+ Equation 12.5.4 */
+
+ for (k = 1; k < GSL_MIN(M,N+1); k++)
+ {
+ double c, s;
+ double diag = gsl_matrix_get (L, k - 1, k - 1);
+ double offdiag = gsl_matrix_get (L, k - 1 , k);
+
+ create_givens (diag, offdiag, &c, &s);
+ apply_givens_lq (M, N, Q, L, k - 1, k, c, s);
+
+ gsl_matrix_set (L, k - 1, k, 0.0); /* exact zero of G^T */
+ }
+
+ return GSL_SUCCESS;
+ }
+}
+
+int
+gsl_linalg_LQ_LQsolve (gsl_matrix * Q, gsl_matrix * L, const gsl_vector * b, gsl_vector * x)
+{
+ const size_t N = L->size1;
+ const size_t M = L->size2;
+
+ if (M != N)
+ {
+ return GSL_ENOTSQR;
+ }
+ else if (Q->size1 != M || b->size != M || x->size != M)
+ {
+ return GSL_EBADLEN;
+ }
+ else
+ {
+ /* compute sol = b^T Q^T */
+
+ gsl_blas_dgemv (CblasNoTrans, 1.0, Q, b, 0.0, x);
+
+ /* Solve x^T L = sol, storing x in-place */
+
+ gsl_blas_dtrsv (CblasLower, CblasTrans, CblasNonUnit, L, x);
+
+ return GSL_SUCCESS;
+ }
+}
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