schur.h

强大的C++库

/*!
 * \file
 * \brief Definitions of Schur decomposition functions
 * \author Adam Piatyszek
 *
 * -------------------------------------------------------------------------
 *
 * IT++ - C++ library of mathematical, signal processing, speech processing,
 *        and communications classes and functions
 *
 * Copyright (C) 1995-2008  (see AUTHORS file for a list of contributors)
 *
 * 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 St, Fifth Floor, Boston, MA 02110-1301 USA
 *
 * -------------------------------------------------------------------------
 */

#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