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+/* linalg/bidiag.c
+ *
+ * Copyright (C) 2001 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.
+ */
+
+/* Factorise a matrix A into
+ *
+ * A = U B V^T
+ *
+ * where U and V are orthogonal and B is upper bidiagonal.
+ *
+ * On exit, B is stored in the diagonal and first superdiagonal of A.
+ *
+ * U is stored as a packed set of Householder transformations in the
+ * lower triangular part of the input matrix below the diagonal.
+ *
+ * V is stored as a packed set of Householder transformations in the
+ * upper triangular part of the input matrix above the first
+ * superdiagonal.
+ *
+ * The full matrix for U can be obtained as the product
+ *
+ * U = U_1 U_2 .. U_N
+ *
+ * where
+ *
+ * U_i = (I - tau_i * u_i * u_i')
+ *
+ * and where u_i is a Householder vector
+ *
+ * u_i = [0, .. , 0, 1, A(i+1,i), A(i+3,i), .. , A(M,i)]
+ *
+ * The full matrix for V can be obtained as the product
+ *
+ * V = V_1 V_2 .. V_(N-2)
+ *
+ * where
+ *
+ * V_i = (I - tau_i * v_i * v_i')
+ *
+ * and where v_i is a Householder vector
+ *
+ * v_i = [0, .. , 0, 1, A(i,i+2), A(i,i+3), .. , A(i,N)]
+ *
+ * See Golub & Van Loan, "Matrix Computations" (3rd ed), Algorithm 5.4.2
+ *
+ * Note: this description uses 1-based indices. The code below uses
+ * 0-based indices
+ */
+
+#include <config.h>
+#include <stdlib.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>
+
+int
+gsl_linalg_bidiag_decomp (gsl_matrix * A, gsl_vector * tau_U, gsl_vector * tau_V)
+{
+ if (A->size1 < A->size2)
+ {
+ GSL_ERROR ("bidiagonal decomposition requires M>=N", GSL_EBADLEN);
+ }
+ else if (tau_U->size != A->size2)
+ {
+ GSL_ERROR ("size of tau_U must be N", GSL_EBADLEN);
+ }
+ else if (tau_V->size + 1 != A->size2)
+ {
+ GSL_ERROR ("size of tau_V must be (N - 1)", GSL_EBADLEN);
+ }
+ else
+ {
+ const size_t M = A->size1;
+ const size_t N = A->size2;
+ size_t i;
+
+ for (i = 0 ; i < N; i++)
+ {
+ /* Apply Householder transformation to current column */
+
+ {
+ gsl_vector_view c = gsl_matrix_column (A, i);
+ gsl_vector_view v = gsl_vector_subvector (&c.vector, i, M - i);
+ double tau_i = gsl_linalg_householder_transform (&v.vector);
+
+ /* 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, &v.vector, &m.matrix);
+ }
+
+ gsl_vector_set (tau_U, i, tau_i);
+
+ }
+
+ /* Apply Householder transformation to current row */
+
+ if (i + 1 < N)
+ {
+ gsl_vector_view r = gsl_matrix_row (A, i);
+ gsl_vector_view v = gsl_vector_subvector (&r.vector, i + 1, N - (i + 1));
+ double tau_i = gsl_linalg_householder_transform (&v.vector);
+
+ /* Apply the transformation to the remaining rows */
+
+ if (i + 1 < M)
+ {
+ gsl_matrix_view m =
+ gsl_matrix_submatrix (A, i+1, i+1, M - (i+1), N - (i+1));
+ gsl_linalg_householder_mh (tau_i, &v.vector, &m.matrix);
+ }
+
+ gsl_vector_set (tau_V, i, tau_i);
+ }
+ }
+ }
+
+ return GSL_SUCCESS;
+}
+
+/* Form the orthogonal matrices U, V, diagonal d and superdiagonal sd
+ from the packed bidiagonal matrix A */
+
+int
+gsl_linalg_bidiag_unpack (const gsl_matrix * A,
+ const gsl_vector * tau_U,
+ gsl_matrix * U,
+ const gsl_vector * tau_V,
+ gsl_matrix * V,
+ gsl_vector * diag,
+ gsl_vector * superdiag)
+{
+ const size_t M = A->size1;
+ const size_t N = A->size2;
+
+ const size_t K = GSL_MIN(M, N);
+
+ if (M < N)
+ {
+ GSL_ERROR ("matrix A must have M >= N", GSL_EBADLEN);
+ }
+ else if (tau_U->size != K)
+ {
+ GSL_ERROR ("size of tau must be MIN(M,N)", GSL_EBADLEN);
+ }
+ else if (tau_V->size + 1 != K)
+ {
+ GSL_ERROR ("size of tau must be MIN(M,N) - 1", GSL_EBADLEN);
+ }
+ else if (U->size1 != M || U->size2 != N)
+ {
+ GSL_ERROR ("size of U must be M x N", GSL_EBADLEN);
+ }
+ else if (V->size1 != N || V->size2 != N)
+ {
+ GSL_ERROR ("size of V must be N x N", GSL_EBADLEN);
+ }
+ else if (diag->size != K)
+ {
+ GSL_ERROR ("size of diagonal must match size of A", GSL_EBADLEN);
+ }
+ else if (superdiag->size + 1 != K)
+ {
+ GSL_ERROR ("size of subdiagonal must be (diagonal size - 1)", GSL_EBADLEN);
+ }
+ else
+ {
+ size_t i, j;
+
+ /* Copy diagonal into diag */
+
+ for (i = 0; i < N; i++)
+ {
+ double Aii = gsl_matrix_get (A, i, i);
+ gsl_vector_set (diag, i, Aii);
+ }
+
+ /* Copy superdiagonal into superdiag */
+
+ for (i = 0; i < N - 1; i++)
+ {
+ double Aij = gsl_matrix_get (A, i, i+1);
+ gsl_vector_set (superdiag, i, Aij);
+ }
+
+ /* Initialize V to the identity */
+
+ gsl_matrix_set_identity (V);
+
+ for (i = N - 1; i > 0 && i--;)
+ {
+ /* Householder row transformation to accumulate V */
+ gsl_vector_const_view r = gsl_matrix_const_row (A, i);
+ gsl_vector_const_view h =
+ gsl_vector_const_subvector (&r.vector, i + 1, N - (i+1));
+
+ double ti = gsl_vector_get (tau_V, i);
+
+ gsl_matrix_view m =
+ gsl_matrix_submatrix (V, i + 1, i + 1, N-(i+1), N-(i+1));
+
+ gsl_linalg_householder_hm (ti, &h.vector, &m.matrix);
+ }
+
+ /* Initialize U to the identity */
+
+ gsl_matrix_set_identity (U);
+
+ for (j = N; j > 0 && j--;)
+ {
+ /* Householder column transformation to accumulate U */
+ gsl_vector_const_view c = gsl_matrix_const_column (A, j);
+ gsl_vector_const_view h = gsl_vector_const_subvector (&c.vector, j, M - j);
+ double tj = gsl_vector_get (tau_U, j);
+
+ gsl_matrix_view m =
+ gsl_matrix_submatrix (U, j, j, M-j, N-j);
+
+ gsl_linalg_householder_hm (tj, &h.vector, &m.matrix);
+ }
+
+ return GSL_SUCCESS;
+ }
+}
+
+int
+gsl_linalg_bidiag_unpack2 (gsl_matrix * A,
+ gsl_vector * tau_U,
+ gsl_vector * tau_V,
+ gsl_matrix * V)
+{
+ const size_t M = A->size1;
+ const size_t N = A->size2;
+
+ const size_t K = GSL_MIN(M, N);
+
+ if (M < N)
+ {
+ GSL_ERROR ("matrix A must have M >= N", GSL_EBADLEN);
+ }
+ else if (tau_U->size != K)
+ {
+ GSL_ERROR ("size of tau must be MIN(M,N)", GSL_EBADLEN);
+ }
+ else if (tau_V->size + 1 != K)
+ {
+ GSL_ERROR ("size of tau must be MIN(M,N) - 1", GSL_EBADLEN);
+ }
+ else if (V->size1 != N || V->size2 != N)
+ {
+ GSL_ERROR ("size of V must be N x N", GSL_EBADLEN);
+ }
+ else
+ {
+ size_t i, j;
+
+ /* Initialize V to the identity */
+
+ gsl_matrix_set_identity (V);
+
+ for (i = N - 1; i > 0 && i--;)
+ {
+ /* Householder row transformation to accumulate V */
+ gsl_vector_const_view r = gsl_matrix_const_row (A, i);
+ gsl_vector_const_view h =
+ gsl_vector_const_subvector (&r.vector, i + 1, N - (i+1));
+
+ double ti = gsl_vector_get (tau_V, i);
+
+ gsl_matrix_view m =
+ gsl_matrix_submatrix (V, i + 1, i + 1, N-(i+1), N-(i+1));
+
+ gsl_linalg_householder_hm (ti, &h.vector, &m.matrix);
+ }
+
+ /* Copy superdiagonal into tau_v */
+
+ for (i = 0; i < N - 1; i++)
+ {
+ double Aij = gsl_matrix_get (A, i, i+1);
+ gsl_vector_set (tau_V, i, Aij);
+ }
+
+ /* Allow U to be unpacked into the same memory as A, copy
+ diagonal into tau_U */
+
+ for (j = N; j > 0 && j--;)
+ {
+ /* Householder column transformation to accumulate U */
+ double tj = gsl_vector_get (tau_U, j);
+ double Ajj = gsl_matrix_get (A, j, j);
+ gsl_matrix_view m = gsl_matrix_submatrix (A, j, j, M-j, N-j);
+
+ gsl_vector_set (tau_U, j, Ajj);
+ gsl_linalg_householder_hm1 (tj, &m.matrix);
+ }
+
+ return GSL_SUCCESS;
+ }
+}
+
+
+int
+gsl_linalg_bidiag_unpack_B (const gsl_matrix * A,
+ gsl_vector * diag,
+ gsl_vector * superdiag)
+{
+ const size_t M = A->size1;
+ const size_t N = A->size2;
+
+ const size_t K = GSL_MIN(M, N);
+
+ if (diag->size != K)
+ {
+ GSL_ERROR ("size of diagonal must match size of A", GSL_EBADLEN);
+ }
+ else if (superdiag->size + 1 != K)
+ {
+ GSL_ERROR ("size of subdiagonal must be (matrix size - 1)", GSL_EBADLEN);
+ }
+ else
+ {
+ size_t i;
+
+ /* Copy diagonal into diag */
+
+ for (i = 0; i < K; i++)
+ {
+ double Aii = gsl_matrix_get (A, i, i);
+ gsl_vector_set (diag, i, Aii);
+ }
+
+ /* Copy superdiagonal into superdiag */
+
+ for (i = 0; i < K - 1; i++)
+ {
+ double Aij = gsl_matrix_get (A, i, i+1);
+ gsl_vector_set (superdiag, i, Aij);
+ }
+
+ return GSL_SUCCESS;
+ }
+}
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