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Diffstat (limited to 'gsl-1.9/linalg/qr.c')
-rw-r--r-- | gsl-1.9/linalg/qr.c | 566 |
1 files changed, 566 insertions, 0 deletions
diff --git a/gsl-1.9/linalg/qr.c b/gsl-1.9/linalg/qr.c new file mode 100644 index 0000000..f3526e6 --- /dev/null +++ b/gsl-1.9/linalg/qr.c @@ -0,0 +1,566 @@ +/* linalg/qr.c + * + * Copyright (C) 1996, 1997, 1998, 1999, 2000 Gerard Jungman, 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 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. + */ + +/* Author: G. Jungman */ + +#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" + +/* Factorise a general M x N matrix A into + * + * A = Q R + * + * where Q is orthogonal (M x M) and R is upper triangular (M x N). + * + * Q is stored as a packed set of Householder transformations in the + * strict lower triangular part of the input matrix. + * + * R is stored in the diagonal and upper 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_QR_decomp (gsl_matrix * A, gsl_vector * tau) +{ + const size_t M = A->size1; + const size_t N = 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_column (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, i + 1, M - i, N - (i + 1)); + gsl_linalg_householder_hm (tau_i, &(c.vector), &(m.matrix)); + } + } + + return GSL_SUCCESS; + } +} + +/* Solves the system A x = b using the QR factorisation, + + * R x = Q^T b + * + * to obtain x. Based on SLATEC code. + */ + +int +gsl_linalg_QR_solve (const gsl_matrix * QR, const gsl_vector * tau, const gsl_vector * b, gsl_vector * x) +{ + if (QR->size1 != QR->size2) + { + GSL_ERROR ("QR matrix must be square", GSL_ENOTSQR); + } + else if (QR->size1 != b->size) + { + GSL_ERROR ("matrix size must match b size", GSL_EBADLEN); + } + else if (QR->size2 != 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_QR_svx (QR, tau, x); + + return GSL_SUCCESS; + } +} + +/* Solves the system A x = b in place using the QR factorisation, + + * R x = Q^T b + * + * to obtain x. Based on SLATEC code. + */ + +int +gsl_linalg_QR_svx (const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * x) +{ + + if (QR->size1 != QR->size2) + { + GSL_ERROR ("QR matrix must be square", GSL_ENOTSQR); + } + else if (QR->size1 != x->size) + { + GSL_ERROR ("matrix size must match x/rhs size", GSL_EBADLEN); + } + else + { + /* compute rhs = Q^T b */ + + gsl_linalg_QR_QTvec (QR, tau, x); + + /* Solve R x = rhs, storing x in-place */ + + gsl_blas_dtrsv (CblasUpper, CblasNoTrans, CblasNonUnit, QR, x); + + return GSL_SUCCESS; + } +} + + +/* Find the least squares solution to the overdetermined system + * + * A x = b + * + * for M >= N using the QR factorization A = Q R. + */ + +int +gsl_linalg_QR_lssolve (const gsl_matrix * QR, const gsl_vector * tau, const gsl_vector * b, gsl_vector * x, gsl_vector * residual) +{ + const size_t M = QR->size1; + const size_t N = QR->size2; + + if (M < N) + { + GSL_ERROR ("QR 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 R = gsl_matrix_const_submatrix (QR, 0, 0, N, N); + gsl_vector_view c = gsl_vector_subvector(residual, 0, N); + + gsl_vector_memcpy(residual, b); + + /* compute rhs = Q^T b */ + + gsl_linalg_QR_QTvec (QR, tau, residual); + + /* Solve R x = rhs */ + + gsl_vector_memcpy(x, &(c.vector)); + + gsl_blas_dtrsv (CblasUpper, CblasNoTrans, CblasNonUnit, &(R.matrix), x); + + /* Compute residual = b - A x = Q (Q^T b - R x) */ + + gsl_vector_set_zero(&(c.vector)); + + gsl_linalg_QR_Qvec(QR, tau, residual); + + return GSL_SUCCESS; + } +} + + +int +gsl_linalg_QR_Rsolve (const gsl_matrix * QR, const gsl_vector * b, gsl_vector * x) +{ + if (QR->size1 != QR->size2) + { + GSL_ERROR ("QR matrix must be square", GSL_ENOTSQR); + } + else if (QR->size1 != b->size) + { + GSL_ERROR ("matrix size must match b size", GSL_EBADLEN); + } + else if (QR->size2 != 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 (CblasUpper, CblasNoTrans, CblasNonUnit, QR, x); + + return GSL_SUCCESS; + } +} + + +int +gsl_linalg_QR_Rsvx (const gsl_matrix * QR, gsl_vector * x) +{ + if (QR->size1 != QR->size2) + { + GSL_ERROR ("QR matrix must be square", GSL_ENOTSQR); + } + else if (QR->size1 != x->size) + { + GSL_ERROR ("matrix size must match rhs size", GSL_EBADLEN); + } + else + { + /* Solve R x = b, storing x in-place */ + + gsl_blas_dtrsv (CblasUpper, CblasNoTrans, CblasNonUnit, QR, x); + + return GSL_SUCCESS; + } +} + +int +gsl_linalg_R_solve (const gsl_matrix * R, const gsl_vector * b, gsl_vector * x) +{ + if (R->size1 != R->size2) + { + GSL_ERROR ("R matrix must be square", GSL_ENOTSQR); + } + else if (R->size1 != b->size) + { + GSL_ERROR ("matrix size must match b size", GSL_EBADLEN); + } + else if (R->size2 != 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 (CblasUpper, CblasNoTrans, CblasNonUnit, R, x); + + return GSL_SUCCESS; + } +} + +int +gsl_linalg_R_svx (const gsl_matrix * R, gsl_vector * x) +{ + if (R->size1 != R->size2) + { + GSL_ERROR ("R matrix must be square", GSL_ENOTSQR); + } + else if (R->size2 != x->size) + { + GSL_ERROR ("matrix size must match solution size", GSL_EBADLEN); + } + else + { + /* Solve R x = b, storing x inplace in b */ + + gsl_blas_dtrsv (CblasUpper, CblasNoTrans, CblasNonUnit, R, x); + + return GSL_SUCCESS; + } +} + + + +/* Form the product Q^T v from a QR factorized matrix + */ + +int +gsl_linalg_QR_QTvec (const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * v) +{ + const size_t M = QR->size1; + const size_t N = QR->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 N", GSL_EBADLEN); + } + else + { + size_t i; + + /* compute Q^T v */ + + for (i = 0; i < GSL_MIN (M, N); i++) + { + gsl_vector_const_view c = gsl_matrix_const_column (QR, 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_QR_Qvec (const gsl_matrix * QR, const gsl_vector * tau, gsl_vector * v) +{ + const size_t M = QR->size1; + const size_t N = QR->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 N", GSL_EBADLEN); + } + else + { + size_t i; + + /* compute Q^T v */ + + for (i = GSL_MIN (M, N); i > 0 && i--;) + { + gsl_vector_const_view c = gsl_matrix_const_column (QR, 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 QR matrix */ + +int +gsl_linalg_QR_unpack (const gsl_matrix * QR, const gsl_vector * tau, gsl_matrix * Q, gsl_matrix * R) +{ + const size_t M = QR->size1; + const size_t N = QR->size2; + + if (Q->size1 != M || Q->size2 != M) + { + GSL_ERROR ("Q matrix must be M x M", GSL_ENOTSQR); + } + else if (R->size1 != M || R->size2 != N) + { + GSL_ERROR ("R matrix must be M x N", 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; + + /* 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_column (QR, 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_hm (ti, &h.vector, &m.matrix); + } + + /* Form the right triangular matrix R from a packed QR matrix */ + + for (i = 0; i < M; i++) + { + for (j = 0; j < i && j < N; j++) + gsl_matrix_set (R, i, j, 0.0); + + for (j = i; j < N; j++) + gsl_matrix_set (R, i, j, gsl_matrix_get (QR, i, j)); + } + + return GSL_SUCCESS; + } +} + + +/* Update a QR factorisation for A= Q R , A' = A + u v^T, + + * Q' R' = QR + u v^T + * = Q (R + Q^T u v^T) + * = Q (R + w v^T) + * + * where w = Q^T u. + * + * Algorithm from Golub and Van Loan, "Matrix Computations", Section + * 12.5 (Updating Matrix Factorizations, Rank-One Changes) + */ + +int +gsl_linalg_QR_update (gsl_matrix * Q, gsl_matrix * R, + gsl_vector * w, const gsl_vector * v) +{ + const size_t M = R->size1; + const size_t N = R->size2; + + if (Q->size1 != M || Q->size2 != M) + { + GSL_ERROR ("Q matrix must be M x M if R is M x N", GSL_ENOTSQR); + } + else if (w->size != M) + { + GSL_ERROR ("w must be length M if R is M x N", GSL_EBADLEN); + } + else if (v->size != N) + { + GSL_ERROR ("v must be length N if R 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 R, H = J_1^T ... J^T_(n-1) R + 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_qr (M, N, Q, R, k - 1, k, c, s); + } + + w0 = gsl_vector_get (w, 0); + + /* Add in w v^T (Equation 12.5.3) */ + + for (j = 0; j < N; j++) + { + double r0j = gsl_matrix_get (R, 0, j); + double vj = gsl_vector_get (v, j); + gsl_matrix_set (R, 0, j, r0j + w0 * vj); + } + + /* Apply Givens transformations R' = 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 (R, k - 1, k - 1); + double offdiag = gsl_matrix_get (R, k, k - 1); + + create_givens (diag, offdiag, &c, &s); + apply_givens_qr (M, N, Q, R, k - 1, k, c, s); + + gsl_matrix_set (R, k, k - 1, 0.0); /* exact zero of G^T */ + } + + return GSL_SUCCESS; + } +} + +int +gsl_linalg_QR_QRsolve (gsl_matrix * Q, gsl_matrix * R, const gsl_vector * b, gsl_vector * x) +{ + const size_t M = R->size1; + const size_t N = R->size2; + + if (M != N) + { + return GSL_ENOTSQR; + } + else if (Q->size1 != M || b->size != M || x->size != M) + { + return GSL_EBADLEN; + } + else + { + /* compute sol = Q^T b */ + + gsl_blas_dgemv (CblasTrans, 1.0, Q, b, 0.0, x); + + /* Solve R x = sol, storing x in-place */ + + gsl_blas_dtrsv (CblasUpper, CblasNoTrans, CblasNonUnit, R, x); + + return GSL_SUCCESS; + } +} |