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Diffstat (limited to 'gsl-1.9/linalg/qrpt.c')
-rw-r--r-- | gsl-1.9/linalg/qrpt.c | 486 |
1 files changed, 486 insertions, 0 deletions
diff --git a/gsl-1.9/linalg/qrpt.c b/gsl-1.9/linalg/qrpt.c new file mode 100644 index 0000000..ac38547 --- /dev/null +++ b/gsl-1.9/linalg/qrpt.c @@ -0,0 +1,486 @@ +/* linalg/qrpt.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. + */ + +#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_permute_vector.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 P = Q R + * + * where Q is orthogonal (M x M) and R is upper triangular (M x N). + * When A is rank deficient, r = rank(A) < n, then the permutation is + * used to ensure that the lower n - r rows of R are zero and the first + * r columns of Q form an orthonormal basis for A. + * + * 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. + * + * P: column j of P is column k of the identity matrix, where k = + * permutation->data[j] + * + * 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. See LAPACK's + * dgeqpf.f for details. + * + */ + +int +gsl_linalg_QRPT_decomp (gsl_matrix * A, gsl_vector * tau, gsl_permutation * p, int *signum, gsl_vector * norm) +{ + 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 if (p->size != N) + { + GSL_ERROR ("permutation size must be N", GSL_EBADLEN); + } + else if (norm->size != N) + { + GSL_ERROR ("norm size must be N", GSL_EBADLEN); + } + else + { + size_t i; + + *signum = 1; + + gsl_permutation_init (p); /* set to identity */ + + /* Compute column norms and store in workspace */ + + for (i = 0; i < N; i++) + { + gsl_vector_view c = gsl_matrix_column (A, i); + double x = gsl_blas_dnrm2 (&c.vector); + gsl_vector_set (norm, i, x); + } + + for (i = 0; i < GSL_MIN (M, N); i++) + { + /* Bring the column of largest norm into the pivot position */ + + double max_norm = gsl_vector_get(norm, i); + size_t j, kmax = i; + + for (j = i + 1; j < N; j++) + { + double x = gsl_vector_get (norm, j); + + if (x > max_norm) + { + max_norm = x; + kmax = j; + } + } + + if (kmax != i) + { + gsl_matrix_swap_columns (A, i, kmax); + gsl_permutation_swap (p, i, kmax); + gsl_vector_swap_elements(norm,i,kmax); + + (*signum) = -(*signum); + } + + /* 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 */ + + 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); + } + } + + /* Update the norms of the remaining columns too */ + + if (i + 1 < M) + { + for (j = i + 1; j < N; j++) + { + double x = gsl_vector_get (norm, j); + + if (x > 0.0) + { + double y = 0; + double temp= gsl_matrix_get (A, i, j) / x; + + if (fabs (temp) >= 1) + y = 0.0; + else + y = x * sqrt (1 - temp * temp); + + /* recompute norm to prevent loss of accuracy */ + + if (fabs (y / x) < sqrt (20.0) * GSL_SQRT_DBL_EPSILON) + { + gsl_vector_view c_full = gsl_matrix_column (A, j); + gsl_vector_view c = + gsl_vector_subvector(&c_full.vector, + i+1, M - (i+1)); + y = gsl_blas_dnrm2 (&c.vector); + } + + gsl_vector_set (norm, j, y); + } + } + } + } + + return GSL_SUCCESS; + } +} + +int +gsl_linalg_QRPT_decomp2 (const gsl_matrix * A, gsl_matrix * q, gsl_matrix * r, gsl_vector * tau, gsl_permutation * p, int *signum, gsl_vector * norm) +{ + const size_t M = A->size1; + const size_t N = A->size2; + + if (q->size1 != M || q->size2 !=M) + { + GSL_ERROR ("q must be M x M", GSL_EBADLEN); + } + else if (r->size1 != M || r->size2 !=N) + { + GSL_ERROR ("r must be M x N", GSL_EBADLEN); + } + else if (tau->size != GSL_MIN (M, N)) + { + GSL_ERROR ("size of tau must be MIN(M,N)", GSL_EBADLEN); + } + else if (p->size != N) + { + GSL_ERROR ("permutation size must be N", GSL_EBADLEN); + } + else if (norm->size != N) + { + GSL_ERROR ("norm size must be N", GSL_EBADLEN); + } + + gsl_matrix_memcpy (r, A); + + gsl_linalg_QRPT_decomp (r, tau, p, signum, norm); + + /* FIXME: aliased arguments depends on behavior of unpack routine! */ + + gsl_linalg_QR_unpack (r, tau, q, r); + + return GSL_SUCCESS; +} + + +/* Solves the system A x = b using the Q R P^T factorisation, + + R z = Q^T b + + x = P z; + + to obtain x. Based on SLATEC code. */ + +int +gsl_linalg_QRPT_solve (const gsl_matrix * QR, + const gsl_vector * tau, + const gsl_permutation * p, + 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 != p->size) + { + GSL_ERROR ("matrix size must match permutation size", GSL_EBADLEN); + } + 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 + { + gsl_vector_memcpy (x, b); + + gsl_linalg_QRPT_svx (QR, tau, p, x); + + return GSL_SUCCESS; + } +} + +int +gsl_linalg_QRPT_svx (const gsl_matrix * QR, + const gsl_vector * tau, + const gsl_permutation * p, + gsl_vector * x) +{ + if (QR->size1 != QR->size2) + { + GSL_ERROR ("QR matrix must be square", GSL_ENOTSQR); + } + else if (QR->size1 != p->size) + { + GSL_ERROR ("matrix size must match permutation size", GSL_EBADLEN); + } + else if (QR->size2 != x->size) + { + GSL_ERROR ("matrix size must match solution size", GSL_EBADLEN); + } + else + { + /* compute sol = Q^T b */ + + gsl_linalg_QR_QTvec (QR, tau, x); + + /* Solve R x = sol, storing x inplace in sol */ + + gsl_blas_dtrsv (CblasUpper, CblasNoTrans, CblasNonUnit, QR, x); + + gsl_permute_vector_inverse (p, x); + + return GSL_SUCCESS; + } +} + + +int +gsl_linalg_QRPT_QRsolve (const gsl_matrix * Q, const gsl_matrix * R, + const gsl_permutation * p, + const gsl_vector * b, + gsl_vector * x) +{ + if (Q->size1 != Q->size2 || R->size1 != R->size2) + { + return GSL_ENOTSQR; + } + else if (Q->size1 != p->size || Q->size1 != R->size1 + || Q->size1 != b->size) + { + return GSL_EBADLEN; + } + else + { + /* compute b' = Q^T b */ + + gsl_blas_dgemv (CblasTrans, 1.0, Q, b, 0.0, x); + + /* Solve R x = b', storing x inplace */ + + gsl_blas_dtrsv (CblasUpper, CblasNoTrans, CblasNonUnit, R, x); + + /* Apply permutation to solution in place */ + + gsl_permute_vector_inverse (p, x); + + return GSL_SUCCESS; + } +} + +int +gsl_linalg_QRPT_Rsolve (const gsl_matrix * QR, + const gsl_permutation * p, + 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 if (p->size != x->size) + { + GSL_ERROR ("permutation size must match x size", GSL_EBADLEN); + } + else + { + /* Copy x <- b */ + + gsl_vector_memcpy (x, b); + + /* Solve R x = b, storing x inplace */ + + gsl_blas_dtrsv (CblasUpper, CblasNoTrans, CblasNonUnit, QR, x); + + gsl_permute_vector_inverse (p, x); + + return GSL_SUCCESS; + } +} + + +int +gsl_linalg_QRPT_Rsvx (const gsl_matrix * QR, + const gsl_permutation * p, + gsl_vector * x) +{ + if (QR->size1 != QR->size2) + { + GSL_ERROR ("QR matrix must be square", GSL_ENOTSQR); + } + else if (QR->size2 != x->size) + { + GSL_ERROR ("matrix size must match x size", GSL_EBADLEN); + } + else if (p->size != x->size) + { + GSL_ERROR ("permutation size must match x size", GSL_EBADLEN); + } + else + { + /* Solve R x = b, storing x inplace */ + + gsl_blas_dtrsv (CblasUpper, CblasNoTrans, CblasNonUnit, QR, x); + + gsl_permute_vector_inverse (p, x); + + return GSL_SUCCESS; + } +} + + + +/* Update a Q R P^T factorisation for A P= Q R , A' = A + u v^T, + + Q' R' P^-1 = QR P^-1 + u v^T + = Q (R + Q^T u v^T P ) P^-1 + = Q (R + w v^T P) P^-1 + + 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_QRPT_update (gsl_matrix * Q, gsl_matrix * R, + const gsl_permutation * p, + gsl_vector * w, const gsl_vector * v) +{ + if (Q->size1 != Q->size2 || R->size1 != R->size2) + { + return GSL_ENOTSQR; + } + else if (R->size1 != Q->size2 || v->size != Q->size2 || w->size != Q->size2) + { + return GSL_EBADLEN; + } + else + { + size_t j, k; + const size_t M = Q->size1; + const size_t N = Q->size2; + 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 = N - 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); + size_t p_j = gsl_permutation_get (p, j); + double vj = gsl_vector_get (v, p_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 < N; 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); + } + + return GSL_SUCCESS; + } +} |