| /* 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; |
| } |
| } |