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/* linalg/choleskyc.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 <config.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_linalg.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_complex.h>
#include <gsl/gsl_complex_math.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_errno.h>
/*
* This module contains routines related to the Cholesky decomposition
* of a complex Hermitian positive definite matrix.
*/
static void cholesky_complex_conj_vector(gsl_vector_complex *v);
/*
gsl_linalg_complex_cholesky_decomp()
Perform the Cholesky decomposition on a Hermitian positive definite
matrix. See Golub & Van Loan, "Matrix Computations" (3rd ed),
algorithm 4.2.2.
Inputs: A - (input/output) complex postive definite matrix
Return: success or error
The lower triangle of A is overwritten with the Cholesky decomposition
*/
int
gsl_linalg_complex_cholesky_decomp(gsl_matrix_complex *A)
{
const size_t N = A->size1;
if (N != A->size2)
{
GSL_ERROR("cholesky decomposition requires square matrix", GSL_ENOTSQR);
}
else
{
size_t i, j;
gsl_complex z;
double ajj;
for (j = 0; j < N; ++j)
{
z = gsl_matrix_complex_get(A, j, j);
ajj = GSL_REAL(z);
if (j > 0)
{
gsl_vector_complex_const_view aj =
gsl_matrix_complex_const_subrow(A, j, 0, j);
gsl_blas_zdotc(&aj.vector, &aj.vector, &z);
ajj -= GSL_REAL(z);
}
if (ajj <= 0.0)
{
GSL_ERROR("matrix is not positive definite", GSL_EDOM);
}
ajj = sqrt(ajj);
GSL_SET_COMPLEX(&z, ajj, 0.0);
gsl_matrix_complex_set(A, j, j, z);
if (j < N - 1)
{
gsl_vector_complex_view av =
gsl_matrix_complex_subcolumn(A, j, j + 1, N - j - 1);
if (j > 0)
{
gsl_vector_complex_view aj =
gsl_matrix_complex_subrow(A, j, 0, j);
gsl_matrix_complex_view am =
gsl_matrix_complex_submatrix(A, j + 1, 0, N - j - 1, j);
cholesky_complex_conj_vector(&aj.vector);
gsl_blas_zgemv(CblasNoTrans,
GSL_COMPLEX_NEGONE,
&am.matrix,
&aj.vector,
GSL_COMPLEX_ONE,
&av.vector);
cholesky_complex_conj_vector(&aj.vector);
}
gsl_blas_zdscal(1.0 / ajj, &av.vector);
}
}
/* Now store L^H in upper triangle */
for (i = 1; i < N; ++i)
{
for (j = 0; j < i; ++j)
{
z = gsl_matrix_complex_get(A, i, j);
gsl_matrix_complex_set(A, j, i, gsl_complex_conjugate(z));
}
}
return GSL_SUCCESS;
}
} /* gsl_linalg_complex_cholesky_decomp() */
/*
gsl_linalg_complex_cholesky_solve()
Solve A x = b where A is in cholesky form
*/
int
gsl_linalg_complex_cholesky_solve (const gsl_matrix_complex * cholesky,
const gsl_vector_complex * b,
gsl_vector_complex * x)
{
if (cholesky->size1 != cholesky->size2)
{
GSL_ERROR ("cholesky matrix must be square", GSL_ENOTSQR);
}
else if (cholesky->size1 != b->size)
{
GSL_ERROR ("matrix size must match b size", GSL_EBADLEN);
}
else if (cholesky->size2 != x->size)
{
GSL_ERROR ("matrix size must match solution size", GSL_EBADLEN);
}
else
{
gsl_vector_complex_memcpy (x, b);
/* solve for y using forward-substitution, L y = b */
gsl_blas_ztrsv (CblasLower, CblasNoTrans, CblasNonUnit, cholesky, x);
/* perform back-substitution, L^H x = y */
gsl_blas_ztrsv (CblasLower, CblasConjTrans, CblasNonUnit, cholesky, x);
return GSL_SUCCESS;
}
} /* gsl_linalg_complex_cholesky_solve() */
/*
gsl_linalg_complex_cholesky_svx()
Solve A x = b in place where A is in cholesky form
*/
int
gsl_linalg_complex_cholesky_svx (const gsl_matrix_complex * cholesky,
gsl_vector_complex * x)
{
if (cholesky->size1 != cholesky->size2)
{
GSL_ERROR ("cholesky matrix must be square", GSL_ENOTSQR);
}
else if (cholesky->size2 != x->size)
{
GSL_ERROR ("matrix size must match solution size", GSL_EBADLEN);
}
else
{
/* solve for y using forward-substitution, L y = b */
gsl_blas_ztrsv (CblasLower, CblasNoTrans, CblasNonUnit, cholesky, x);
/* perform back-substitution, L^H x = y */
gsl_blas_ztrsv (CblasLower, CblasConjTrans, CblasNonUnit, cholesky, x);
return GSL_SUCCESS;
}
} /* gsl_linalg_complex_cholesky_svx() */
/********************************************
* INTERNAL ROUTINES *
********************************************/
static void
cholesky_complex_conj_vector(gsl_vector_complex *v)
{
size_t i;
for (i = 0; i < v->size; ++i)
{
gsl_complex z = gsl_vector_complex_get(v, i);
gsl_vector_complex_set(v, i, gsl_complex_conjugate(z));
}
} /* cholesky_complex_conj_vector() */
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