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/* eigen/genherm.c
*
* Copyright (C) 2007 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 3 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 <stdlib.h>
#include <config.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_matrix.h>
#include <gsl/gsl_complex.h>
#include <gsl/gsl_complex_math.h>
/*
* This module computes the eigenvalues of a complex generalized
* hermitian-definite eigensystem A x = \lambda B x, where A and
* B are hermitian, and B is positive-definite.
*/
/*
gsl_eigen_genherm_alloc()
Allocate a workspace for solving the generalized hermitian-definite
eigenvalue problem. The size of this workspace is O(3n).
Inputs: n - size of matrices
Return: pointer to workspace
*/
gsl_eigen_genherm_workspace *
gsl_eigen_genherm_alloc(const size_t n)
{
gsl_eigen_genherm_workspace *w;
if (n == 0)
{
GSL_ERROR_NULL ("matrix dimension must be positive integer",
GSL_EINVAL);
}
w = (gsl_eigen_genherm_workspace *) calloc (1, sizeof (gsl_eigen_genherm_workspace));
if (w == 0)
{
GSL_ERROR_NULL ("failed to allocate space for workspace", GSL_ENOMEM);
}
w->size = n;
w->herm_workspace_p = gsl_eigen_herm_alloc(n);
if (!w->herm_workspace_p)
{
gsl_eigen_genherm_free(w);
GSL_ERROR_NULL("failed to allocate space for herm workspace", GSL_ENOMEM);
}
return (w);
} /* gsl_eigen_genherm_alloc() */
/*
gsl_eigen_genherm_free()
Free workspace w
*/
void
gsl_eigen_genherm_free (gsl_eigen_genherm_workspace * w)
{
RETURN_IF_NULL (w);
if (w->herm_workspace_p)
gsl_eigen_herm_free(w->herm_workspace_p);
free(w);
} /* gsl_eigen_genherm_free() */
/*
gsl_eigen_genherm()
Solve the generalized hermitian-definite eigenvalue problem
A x = \lambda B x
for the eigenvalues \lambda.
Inputs: A - complex hermitian matrix
B - complex hermitian and positive definite matrix
eval - where to store eigenvalues
w - workspace
Return: success or error
*/
int
gsl_eigen_genherm (gsl_matrix_complex * A, gsl_matrix_complex * B,
gsl_vector * eval, gsl_eigen_genherm_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 ((N != B->size1) || (N != B->size2))
{
GSL_ERROR ("B matrix dimensions must match A", GSL_EBADLEN);
}
else if (eval->size != N)
{
GSL_ERROR ("eigenvalue vector must match matrix size", GSL_EBADLEN);
}
else if (w->size != N)
{
GSL_ERROR ("matrix size does not match workspace", GSL_EBADLEN);
}
else
{
int s;
/* compute Cholesky factorization of B */
s = gsl_linalg_complex_cholesky_decomp(B);
if (s != GSL_SUCCESS)
return s; /* B is not positive definite */
/* transform to standard hermitian eigenvalue problem */
gsl_eigen_genherm_standardize(A, B);
s = gsl_eigen_herm(A, eval, w->herm_workspace_p);
return s;
}
} /* gsl_eigen_genherm() */
/*
gsl_eigen_genherm_standardize()
Reduce the generalized hermitian-definite eigenproblem to
the standard hermitian eigenproblem by computing
C = L^{-1} A L^{-H}
where L L^H is the Cholesky decomposition of B
Inputs: A - (input/output) complex hermitian matrix
B - complex hermitian, positive definite matrix in Cholesky form
Return: success
Notes: A is overwritten by L^{-1} A L^{-H}
*/
int
gsl_eigen_genherm_standardize(gsl_matrix_complex *A,
const gsl_matrix_complex *B)
{
const size_t N = A->size1;
size_t i;
double a, b;
gsl_complex y, z;
GSL_SET_IMAG(&z, 0.0);
for (i = 0; i < N; ++i)
{
/* update lower triangle of A(i:n, i:n) */
y = gsl_matrix_complex_get(A, i, i);
a = GSL_REAL(y);
y = gsl_matrix_complex_get(B, i, i);
b = GSL_REAL(y);
a /= b * b;
GSL_SET_REAL(&z, a);
gsl_matrix_complex_set(A, i, i, z);
if (i < N - 1)
{
gsl_vector_complex_view ai =
gsl_matrix_complex_subcolumn(A, i, i + 1, N - i - 1);
gsl_matrix_complex_view ma =
gsl_matrix_complex_submatrix(A, i + 1, i + 1, N - i - 1, N - i - 1);
gsl_vector_complex_const_view bi =
gsl_matrix_complex_const_subcolumn(B, i, i + 1, N - i - 1);
gsl_matrix_complex_const_view mb =
gsl_matrix_complex_const_submatrix(B, i + 1, i + 1, N - i - 1, N - i - 1);
gsl_blas_zdscal(1.0 / b, &ai.vector);
GSL_SET_REAL(&z, -0.5 * a);
gsl_blas_zaxpy(z, &bi.vector, &ai.vector);
gsl_blas_zher2(CblasLower,
GSL_COMPLEX_NEGONE,
&ai.vector,
&bi.vector,
&ma.matrix);
gsl_blas_zaxpy(z, &bi.vector, &ai.vector);
gsl_blas_ztrsv(CblasLower,
CblasNoTrans,
CblasNonUnit,
&mb.matrix,
&ai.vector);
}
}
return GSL_SUCCESS;
} /* gsl_eigen_genherm_standardize() */
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