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/*!
* \file
* \brief Definitions of Schur decomposition functions
* \author Adam Piatyszek
*
* -------------------------------------------------------------------------
*
* Copyright (C) 1995-2008 (see AUTHORS file for a list of contributors)
*
* This file is part of IT++ - a C++ library of mathematical, signal
* processing, speech processing, and communications classes and functions.
*
* IT++ 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.
*
* IT++ 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 IT++. If not, see <http://www.gnu.org/licenses/>.
*
* -------------------------------------------------------------------------
*/
#ifndef SCHUR_H
#define SCHUR_H
#include <itpp/base/mat.h>
namespace itpp {
/*!
* \ingroup matrixdecomp
* \brief Schur decomposition of a real matrix
*
* This function computes the Schur form of a square real matrix
* \f$ \mathbf{A} \f$. The Schur decomposition satisfies the
* following equation:
* \f[ \mathbf{U} \mathbf{T} \mathbf{U}^{T} = \mathbf{A} \f]
* where: \f$ \mathbf{U} \f$ is a unitary, \f$ \mathbf{T} \f$ is upper
* quasi-triangular, and \f$ \mathbf{U}^{T} \f$ is the transposed
* \f$ \mathbf{U} \f$ matrix.
*
* The upper quasi-triangular matrix may have \f$ 2 \times 2 \f$ blocks on
* its diagonal.
*
* Uses the LAPACK routine DGEES.
*/
bool schur(const mat &A, mat &U, mat &T);
/*!
* \ingroup matrixdecomp
* \brief Schur decomposition of a real matrix
*
* This function computes the Schur form of a square real matrix
* \f$ \mathbf{A} \f$. The Schur decomposition satisfies the
* following equation:
* \f[ \mathbf{U} \mathbf{T} \mathbf{U}^{T} = \mathbf{A} \f]
* where: \f$ \mathbf{U} \f$ is a unitary, \f$ \mathbf{T} \f$ is upper
* quasi-triangular, and \f$ \mathbf{U}^{T} \f$ is the transposed
* \f$ \mathbf{U} \f$ matrix.
*
* The upper quasi-triangular matrix may have \f$ 2 \times 2 \f$ blocks on
* its diagonal.
*
* \return Real Schur matrix \f$ \mathbf{T} \f$
*
* uses the LAPACK routine DGEES.
*/
mat schur(const mat &A);
/*!
* \ingroup matrixdecomp
* \brief Schur decomposition of a complex matrix
*
* This function computes the Schur form of a square complex matrix
* \f$ \mathbf{A} \f$. The Schur decomposition satisfies
* the following equation:
* \f[ \mathbf{U} \mathbf{T} \mathbf{U}^{H} = \mathbf{A} \f]
* where: \f$ \mathbf{U} \f$ is a unitary, \f$ \mathbf{T} \f$ is upper
* triangular, and \f$ \mathbf{U}^{H} \f$ is the Hermitian
* transposition of the \f$ \mathbf{U} \f$ matrix.
*
* Uses the LAPACK routine ZGEES.
*/
bool schur(const cmat &A, cmat &U, cmat &T);
/*!
* \ingroup matrixdecomp
* \brief Schur decomposition of a complex matrix
*
* This function computes the Schur form of a square complex matrix
* \f$ \mathbf{A} \f$. The Schur decomposition satisfies
* the following equation:
* \f[ \mathbf{U} \mathbf{T} \mathbf{U}^{H} = \mathbf{A} \f]
* where: \f$ \mathbf{U} \f$ is a unitary, \f$ \mathbf{T} \f$ is upper
* triangular, and \f$ \mathbf{U}^{H} \f$ is the Hermitian
* transposition of the \f$ \mathbf{U} \f$ matrix.
*
* \return Complex Schur matrix \f$ \mathbf{T} \f$
*
* Uses the LAPACK routine ZGEES.
*/
cmat schur(const cmat &A);
} // namespace itpp
#endif // #ifndef SCHUR_H