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+/* eigen/nonsymm.c
+ *
+ * Copyright (C) 2006 Patrick Alken
+ *
+ * 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 <math.h>
+#include <gsl/gsl_eigen.h>
+#include <gsl/gsl_linalg.h>
+#include <gsl/gsl_math.h>
+#include <gsl/gsl_blas.h>
+#include <gsl/gsl_vector.h>
+#include <gsl/gsl_vector_complex.h>
+#include <gsl/gsl_matrix.h>
+
+/*
+ * This module computes the eigenvalues of a real nonsymmetric
+ * matrix, using the double shift Francis method.
+ *
+ * See the references in francis.c.
+ *
+ * This module gets the matrix ready by balancing it and
+ * reducing it to Hessenberg form before passing it to the
+ * francis module.
+ */
+
+/*
+gsl_eigen_nonsymm_alloc()
+
+Allocate a workspace for solving the nonsymmetric eigenvalue problem.
+The size of this workspace is O(2n)
+
+Inputs: n - size of matrix
+
+Return: pointer to workspace
+*/
+
+gsl_eigen_nonsymm_workspace *
+gsl_eigen_nonsymm_alloc(const size_t n)
+{
+ gsl_eigen_nonsymm_workspace *w;
+
+ if (n == 0)
+ {
+ GSL_ERROR_NULL ("matrix dimension must be positive integer",
+ GSL_EINVAL);
+ }
+
+ w = (gsl_eigen_nonsymm_workspace *)
+ malloc (sizeof (gsl_eigen_nonsymm_workspace));
+
+ if (w == 0)
+ {
+ GSL_ERROR_NULL ("failed to allocate space for workspace", GSL_ENOMEM);
+ }
+
+ w->size = n;
+ w->Z = NULL;
+ w->do_balance = 0;
+
+ w->diag = gsl_vector_alloc(n);
+
+ if (w->diag == 0)
+ {
+ GSL_ERROR_NULL ("failed to allocate space for balancing vector", GSL_ENOMEM);
+ }
+
+ w->tau = gsl_vector_alloc(n);
+
+ if (w->tau == 0)
+ {
+ GSL_ERROR_NULL ("failed to allocate space for hessenberg coefficients", GSL_ENOMEM);
+ }
+
+ w->francis_workspace_p = gsl_eigen_francis_alloc();
+
+ if (w->francis_workspace_p == 0)
+ {
+ GSL_ERROR_NULL ("failed to allocate space for francis workspace", GSL_ENOMEM);
+ }
+
+ return (w);
+} /* gsl_eigen_nonsymm_alloc() */
+
+/*
+gsl_eigen_nonsymm_free()
+ Free workspace w
+*/
+
+void
+gsl_eigen_nonsymm_free (gsl_eigen_nonsymm_workspace * w)
+{
+ gsl_vector_free(w->tau);
+
+ gsl_vector_free(w->diag);
+
+ gsl_eigen_francis_free(w->francis_workspace_p);
+
+ free(w);
+} /* gsl_eigen_nonsymm_free() */
+
+/*
+gsl_eigen_nonsymm_params()
+ Set some parameters which define how we solve the eigenvalue
+problem.
+
+Inputs: compute_t - 1 if we want to compute T, 0 if not
+ balance - 1 if we want to balance the matrix, 0 if not
+ w - nonsymm workspace
+*/
+
+void
+gsl_eigen_nonsymm_params (const int compute_t, const int balance,
+ gsl_eigen_nonsymm_workspace *w)
+{
+ gsl_eigen_francis_T(compute_t, w->francis_workspace_p);
+ w->do_balance = balance;
+} /* gsl_eigen_nonsymm_params() */
+
+/*
+gsl_eigen_nonsymm()
+
+Solve the nonsymmetric eigenvalue problem
+
+A x = \lambda x
+
+for the eigenvalues \lambda using the Francis method.
+
+Here we compute the real Schur form
+
+T = Z^t A Z
+
+with the diagonal blocks of T giving us the eigenvalues.
+Z is a matrix of Schur vectors which is not computed by
+this algorithm. See gsl_eigen_nonsymm_Z().
+
+Inputs: A - general real matrix
+ eval - where to store eigenvalues
+ w - workspace
+
+Return: success or error
+
+Notes: If T is computed, it is stored in A on output. Otherwise
+ the diagonal of A contains the 1-by-1 and 2-by-2 eigenvalue
+ blocks.
+*/
+
+int
+gsl_eigen_nonsymm (gsl_matrix * A, gsl_vector_complex * eval,
+ gsl_eigen_nonsymm_workspace * w)
+{
+ const size_t N = A->size1;
+
+ /* check matrix and vector sizes */
+
+ if (N != A->size2)
+ {
+ GSL_ERROR ("matrix must be square to compute eigenvalues", GSL_ENOTSQR);
+ }
+ else if (eval->size != N)
+ {
+ GSL_ERROR ("eigenvalue vector must match matrix size", GSL_EBADLEN);
+ }
+ else
+ {
+ int s;
+
+ if (w->do_balance)
+ {
+ /* balance the matrix */
+ gsl_linalg_balance_matrix(A, w->diag);
+ }
+
+ /* compute the Hessenberg reduction of A */
+ gsl_linalg_hessenberg(A, w->tau);
+
+ if (w->Z)
+ {
+ /*
+ * initialize the matrix Z to U, which is the matrix used
+ * to construct the Hessenberg reduction.
+ */
+
+ /* compute U and store it in Z */
+ gsl_linalg_hessenberg_unpack(A, w->tau, w->Z);
+
+ /* find the eigenvalues and Schur vectors */
+ s = gsl_eigen_francis_Z(A, eval, w->Z, w->francis_workspace_p);
+
+ if (w->do_balance)
+ {
+ /*
+ * The Schur vectors in Z are the vectors for the balanced
+ * matrix. We now must undo the balancing to get the
+ * vectors for the original matrix A.
+ */
+ gsl_linalg_balance_accum(w->Z, w->diag);
+ }
+ }
+ else
+ {
+ /* find the eigenvalues only */
+ s = gsl_eigen_francis(A, eval, w->francis_workspace_p);
+ }
+
+ w->n_evals = w->francis_workspace_p->n_evals;
+
+ return s;
+ }
+} /* gsl_eigen_nonsymm() */
+
+/*
+gsl_eigen_nonsymm_Z()
+
+Solve the nonsymmetric eigenvalue problem
+
+A x = \lambda x
+
+for the eigenvalues \lambda.
+
+Here we compute the real Schur form
+
+T = Z^t A Z
+
+with the diagonal blocks of T giving us the eigenvalues.
+Z is the matrix of Schur vectors.
+
+Inputs: A - general real matrix
+ eval - where to store eigenvalues
+ Z - where to store Schur vectors
+ w - workspace
+
+Return: success or error
+
+Notes: If T is computed, it is stored in A on output. Otherwise
+ the diagonal of A contains the 1-by-1 and 2-by-2 eigenvalue
+ blocks.
+*/
+
+int
+gsl_eigen_nonsymm_Z (gsl_matrix * A, gsl_vector_complex * eval,
+ gsl_matrix * Z, gsl_eigen_nonsymm_workspace * w)
+{
+ /* check matrix and vector sizes */
+
+ if (A->size1 != A->size2)
+ {
+ GSL_ERROR ("matrix must be square to compute eigenvalues", GSL_ENOTSQR);
+ }
+ else if (eval->size != A->size1)
+ {
+ GSL_ERROR ("eigenvalue vector must match matrix size", GSL_EBADLEN);
+ }
+ else if ((Z->size1 != Z->size2) || (Z->size1 != A->size1))
+ {
+ GSL_ERROR ("Z matrix has wrong dimensions", GSL_EBADLEN);
+ }
+ else
+ {
+ int s;
+
+ w->Z = Z;
+
+ s = gsl_eigen_nonsymm(A, eval, w);
+
+ w->Z = NULL;
+
+ return s;
+ }
+} /* gsl_eigen_nonsymm_Z() */