| 1 | /*! |
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| 2 | * \file |
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| 3 | * \brief Definitions of Singular Value Decompositions |
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| 4 | * \author Tony Ottosson |
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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 SVD_H |
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| 31 | #define SVD_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 Get singular values \c s of a real matrix \c A using SVD |
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| 41 | * |
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| 42 | * This function calculates singular values \f$s\f$ from the SVD |
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| 43 | * decomposition of a real matrix \f$A\f$. The SVD algorithm computes the |
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| 44 | * decomposition of a real \f$m \times n\f$ matrix \f$\mathbf{A}\f$ so |
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| 45 | * that |
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| 46 | * \f[ |
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| 47 | * \mathrm{diag}(\mathbf{U}^T \mathbf{A} \mathbf{V}) = \mathbf{s} |
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| 48 | * = \sigma_1, \ldots, \sigma_p |
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| 49 | * \f] |
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| 50 | * where \f$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0\f$ are the |
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| 51 | * singular values of \f$\mathbf{A}\f$. |
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| 52 | * Or put differently: |
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| 53 | * \f[ |
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| 54 | * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^T |
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| 55 | * \f] |
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| 56 | * where \f$ \mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$ |
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| 57 | * |
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| 58 | * \note An external LAPACK library is required by this function. |
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| 59 | */ |
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| 60 | bool svd(const mat &A, vec &s); |
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| 61 | |
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| 62 | /*! |
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| 63 | * \ingroup matrixdecomp |
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| 64 | * \brief Get singular values \c s of a complex matrix \c A using SVD |
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| 65 | * |
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| 66 | * This function calculates singular values \f$s\f$ from the SVD |
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| 67 | * decomposition of a complex matrix \f$A\f$. The SVD algorithm computes |
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| 68 | * the decomposition of a complex \f$m \times n\f$ matrix \f$\mathbf{A}\f$ |
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| 69 | * so that |
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| 70 | * \f[ |
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| 71 | * \mathrm{diag}(\mathbf{U}^H \mathbf{A} \mathbf{V}) = \mathbf{s} |
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| 72 | * = \sigma_1, \ldots, \sigma_p |
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| 73 | * \f] |
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| 74 | * where \f$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0\f$ |
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| 75 | * are the singular values of \f$\mathbf{A}\f$. |
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| 76 | * Or put differently: |
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| 77 | * \f[ |
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| 78 | * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H |
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| 79 | * \f] |
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| 80 | * where \f$ \mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$ |
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| 81 | * |
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| 82 | * \note An external LAPACK library is required by this function. |
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| 83 | */ |
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| 84 | bool svd(const cmat &A, vec &s); |
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| 85 | |
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| 86 | /*! |
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| 87 | * \ingroup matrixdecomp |
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| 88 | * \brief Return singular values of a real matrix \c A using SVD |
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| 89 | * |
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| 90 | * This function returns singular values from the SVD decomposition |
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| 91 | * of a real matrix \f$A\f$. The SVD algorithm computes the decomposition |
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| 92 | * of a real \f$m \times n\f$ matrix \f$\mathbf{A}\f$ so that |
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| 93 | * \f[ |
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| 94 | * \mathrm{diag}(\mathbf{U}^T \mathbf{A} \mathbf{V}) = \mathbf{s} |
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| 95 | * = \sigma_1, \ldots, \sigma_p |
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| 96 | * \f] |
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| 97 | * where \f$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0\f$ are the |
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| 98 | * singular values of \f$\mathbf{A}\f$. |
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| 99 | * Or put differently: |
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| 100 | * \f[ |
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| 101 | * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^T |
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| 102 | * \f] |
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| 103 | * where \f$ \mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$ |
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| 104 | * |
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| 105 | * \note An external LAPACK library is required by this function. |
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| 106 | */ |
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| 107 | vec svd(const mat &A); |
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| 108 | |
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| 109 | /*! |
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| 110 | * \ingroup matrixdecomp |
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| 111 | * \brief Return singular values of a complex matrix \c A using SVD |
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| 112 | * |
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| 113 | * This function returns singular values from the SVD |
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| 114 | * decomposition of a complex matrix \f$A\f$. The SVD algorithm computes |
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| 115 | * the decomposition of a complex \f$m \times n\f$ matrix \f$\mathbf{A}\f$ |
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| 116 | * so that |
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| 117 | * \f[ |
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| 118 | * \mathrm{diag}(\mathbf{U}^H \mathbf{A} \mathbf{V}) = \mathbf{s} |
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| 119 | * = \sigma_1, \ldots, \sigma_p |
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| 120 | * \f] |
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| 121 | * where \f$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0\f$ |
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| 122 | * are the singular values of \f$\mathbf{A}\f$. |
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| 123 | * Or put differently: |
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| 124 | * \f[ |
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| 125 | * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H |
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| 126 | * \f] |
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| 127 | * where \f$ \mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$ |
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| 128 | * |
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| 129 | * \note An external LAPACK library is required by this function. |
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| 130 | */ |
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| 131 | vec svd(const cmat &A); |
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| 132 | |
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| 133 | /*! |
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| 134 | * \ingroup matrixdecomp |
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| 135 | * \brief Perform Singular Value Decomposition (SVD) of a real matrix \c A |
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| 136 | * |
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| 137 | * This function returns two orthonormal matrices \f$U\f$ and \f$V\f$ |
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| 138 | * and a vector of singular values \f$s\f$. |
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| 139 | * The SVD algorithm computes the decomposition of a real \f$m \times n\f$ |
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| 140 | * matrix \f$\mathbf{A}\f$ so that |
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| 141 | * \f[ |
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| 142 | * \mathrm{diag}(\mathbf{U}^T \mathbf{A} \mathbf{V}) = \mathbf{s} |
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| 143 | * = \sigma_1, \ldots, \sigma_p |
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| 144 | * \f] |
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| 145 | * where the elements of \f$\mathbf{s}\f$, \f$\sigma_1 \geq \sigma_2 \geq |
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| 146 | * \ldots \sigma_p \geq 0\f$ are the singular values of \f$\mathbf{A}\f$. |
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| 147 | * Or put differently: |
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| 148 | * \f[ |
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| 149 | * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^T |
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| 150 | * \f] |
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| 151 | * where \f$ \mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$ |
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| 152 | * |
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| 153 | * \note An external LAPACK library is required by this function. |
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| 154 | */ |
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| 155 | bool svd(const mat &A, mat &U, vec &s, mat &V); |
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| 156 | |
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| 157 | /*! |
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| 158 | * \ingroup matrixdecomp |
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| 159 | * \brief Perform Singular Value Decomposition (SVD) of a complex matrix \c A |
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| 160 | * |
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| 161 | * This function returns two orthonormal matrices \f$U\f$ and \f$V\f$ |
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| 162 | * and a vector of singular values \f$s\f$. |
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| 163 | * The SVD algorithm computes the decomposition of a complex \f$m \times n\f$ |
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| 164 | * matrix \f$\mathbf{A}\f$ so that |
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| 165 | * \f[ |
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| 166 | * \mathrm{diag}(\mathbf{U}^H \mathbf{A} \mathbf{V}) = \mathbf{s} |
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| 167 | * = \sigma_1, \ldots, \sigma_p |
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| 168 | * \f] |
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| 169 | * where the elements of \f$\mathbf{s}\f$, \f$\sigma_1 \geq \sigma_2 \geq |
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| 170 | * \ldots \sigma_p \geq 0\f$ are the singular values of \f$\mathbf{A}\f$. |
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| 171 | * Or put differently: |
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| 172 | * \f[ |
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| 173 | * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H |
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| 174 | * \f] |
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| 175 | * where \f$ \mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$ |
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| 176 | * |
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| 177 | * \note An external LAPACK library is required by this function. |
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| 178 | */ |
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| 179 | bool svd(const cmat &A, cmat &U, vec &s, cmat &V); |
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| 180 | |
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| 181 | |
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| 182 | } // namespace itpp |
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| 183 | |
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| 184 | #endif // #ifndef SVD_H |
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