[35] | 1 | /*! |
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| 2 | * \file |
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| 3 | * \brief Definitions of Schur decomposition functions |
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| 4 | * \author Adam Piatyszek |
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| 5 | * |
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| 6 | * ------------------------------------------------------------------------- |
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| 7 | * |
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| 8 | * IT++ - C++ library of mathematical, signal processing, speech processing, |
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| 9 | * and communications classes and functions |
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| 10 | * |
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| 11 | * Copyright (C) 1995-2007 (see AUTHORS file for a list of contributors) |
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| 12 | * |
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| 13 | * This program is free software; you can redistribute it and/or modify |
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| 14 | * it under the terms of the GNU General Public License as published by |
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| 15 | * the Free Software Foundation; either version 2 of the License, or |
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| 16 | * (at your option) any later version. |
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| 17 | * |
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| 18 | * This program is distributed in the hope that it will be useful, |
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| 19 | * but WITHOUT ANY WARRANTY; without even the implied warranty of |
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| 20 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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| 21 | * GNU General Public License for more details. |
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| 22 | * |
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| 23 | * You should have received a copy of the GNU General Public License |
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| 24 | * along with this program; if not, write to the Free Software |
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| 25 | * Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA |
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| 26 | * |
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| 27 | * ------------------------------------------------------------------------- |
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| 28 | */ |
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| 29 | |
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| 30 | #ifndef SCHUR_H |
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| 31 | #define SCHUR_H |
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| 32 | |
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| 33 | #include <itpp/base/mat.h> |
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| 34 | |
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| 35 | |
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| 36 | namespace itpp { |
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| 37 | |
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| 38 | /*! |
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| 39 | * \ingroup matrixdecomp |
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| 40 | * \brief Schur decomposition of a real matrix |
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| 41 | * |
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| 42 | * This function computes the Schur form of a square real matrix |
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| 43 | * \f$ \mathbf{A} \f$. The Schur decomposition satisfies the |
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| 44 | * following equation: |
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| 45 | * \f[ \mathbf{U} \mathbf{T} \mathbf{U}^{T} = \mathbf{A} \f] |
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| 46 | * where: \f$ \mathbf{U} \f$ is a unitary, \f$ \mathbf{T} \f$ is upper |
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| 47 | * quasi-triangular, and \f$ \mathbf{U}^{T} \f$ is the transposed |
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| 48 | * \f$ \mathbf{U} \f$ matrix. |
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| 49 | * |
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| 50 | * The upper quasi-triangular matrix may have \f$ 2 \times 2 \f$ blocks on |
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| 51 | * its diagonal. |
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| 52 | * |
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| 53 | * Uses the LAPACK routine DGEES. |
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| 54 | */ |
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| 55 | bool schur(const mat &A, mat &U, mat &T); |
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| 56 | |
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| 57 | /*! |
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| 58 | * \ingroup matrixdecomp |
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| 59 | * \brief Schur decomposition of a real matrix |
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| 60 | * |
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| 61 | * This function computes the Schur form of a square real matrix |
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| 62 | * \f$ \mathbf{A} \f$. The Schur decomposition satisfies the |
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| 63 | * following equation: |
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| 64 | * \f[ \mathbf{U} \mathbf{T} \mathbf{U}^{T} = \mathbf{A} \f] |
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| 65 | * where: \f$ \mathbf{U} \f$ is a unitary, \f$ \mathbf{T} \f$ is upper |
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| 66 | * quasi-triangular, and \f$ \mathbf{U}^{T} \f$ is the transposed |
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| 67 | * \f$ \mathbf{U} \f$ matrix. |
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| 68 | * |
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| 69 | * The upper quasi-triangular matrix may have \f$ 2 \times 2 \f$ blocks on |
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| 70 | * its diagonal. |
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| 71 | * |
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| 72 | * \return Real Schur matrix \f$ \mathbf{T} \f$ |
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| 73 | * |
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| 74 | * uses the LAPACK routine DGEES. |
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| 75 | */ |
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| 76 | mat schur(const mat &A); |
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| 77 | |
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| 78 | |
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| 79 | /*! |
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| 80 | * \ingroup matrixdecomp |
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| 81 | * \brief Schur decomposition of a complex matrix |
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| 82 | * |
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| 83 | * This function computes the Schur form of a square complex matrix |
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| 84 | * \f$ \mathbf{A} \f$. The Schur decomposition satisfies |
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| 85 | * the following equation: |
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| 86 | * \f[ \mathbf{U} \mathbf{T} \mathbf{U}^{H} = \mathbf{A} \f] |
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| 87 | * where: \f$ \mathbf{U} \f$ is a unitary, \f$ \mathbf{T} \f$ is upper |
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| 88 | * triangular, and \f$ \mathbf{U}^{H} \f$ is the Hermitian |
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| 89 | * transposition of the \f$ \mathbf{U} \f$ matrix. |
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| 90 | * |
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| 91 | * Uses the LAPACK routine ZGEES. |
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| 92 | */ |
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| 93 | bool schur(const cmat &A, cmat &U, cmat &T); |
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| 94 | |
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| 95 | /*! |
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| 96 | * \ingroup matrixdecomp |
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| 97 | * \brief Schur decomposition of a complex matrix |
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| 98 | * |
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| 99 | * This function computes the Schur form of a square complex matrix |
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| 100 | * \f$ \mathbf{A} \f$. The Schur decomposition satisfies |
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| 101 | * the following equation: |
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| 102 | * \f[ \mathbf{U} \mathbf{T} \mathbf{U}^{H} = \mathbf{A} \f] |
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| 103 | * where: \f$ \mathbf{U} \f$ is a unitary, \f$ \mathbf{T} \f$ is upper |
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| 104 | * triangular, and \f$ \mathbf{U}^{H} \f$ is the Hermitian |
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| 105 | * transposition of the \f$ \mathbf{U} \f$ matrix. |
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| 106 | * |
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| 107 | * \return Complex Schur matrix \f$ \mathbf{T} \f$ |
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| 108 | * |
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| 109 | * Uses the LAPACK routine ZGEES. |
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| 110 | */ |
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| 111 | cmat schur(const cmat &A); |
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| 112 | |
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| 113 | |
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| 114 | } // namespace itpp |
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| 115 | |
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| 116 | #endif // #ifndef SCHUR_H |
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