[4ede1e]: itpp / base / algebra / svd.h Maximize Restore History

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 /*! * \file * \brief Definitions of Singular Value Decompositions * \author Tony Ottosson * * ------------------------------------------------------------------------- * * Copyright (C) 1995-2010 (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 . * * ------------------------------------------------------------------------- */ #ifndef SVD_H #define SVD_H #include namespace itpp { /*! * \ingroup matrixdecomp * \brief Get singular values \c s of a real matrix \c A using SVD * * This function calculates singular values \f$s\f$ from the SVD * decomposition of a real matrix \f$A\f$. The SVD algorithm computes the * decomposition of a real \f$m \times n\f$ matrix \f$\mathbf{A}\f$ so * that * \f[ * \mathrm{diag}(\mathbf{U}^T \mathbf{A} \mathbf{V}) = \mathbf{s} * = \sigma_1, \ldots, \sigma_p * \f] * where \f$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0\f$ are the * singular values of \f$\mathbf{A}\f$. * Or put differently: * \f[ * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^T * \f] * where \f$\mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$ * * \note An external LAPACK library is required by this function. */ bool svd(const mat &A, vec &s); /*! * \ingroup matrixdecomp * \brief Get singular values \c s of a complex matrix \c A using SVD * * This function calculates singular values \f$s\f$ from the SVD * decomposition of a complex matrix \f$A\f$. The SVD algorithm computes * the decomposition of a complex \f$m \times n\f$ matrix \f$\mathbf{A}\f$ * so that * \f[ * \mathrm{diag}(\mathbf{U}^H \mathbf{A} \mathbf{V}) = \mathbf{s} * = \sigma_1, \ldots, \sigma_p * \f] * where \f$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0\f$ * are the singular values of \f$\mathbf{A}\f$. * Or put differently: * \f[ * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H * \f] * where \f$\mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$ * * \note An external LAPACK library is required by this function. */ bool svd(const cmat &A, vec &s); /*! * \ingroup matrixdecomp * \brief Return singular values of a real matrix \c A using SVD * * This function returns singular values from the SVD decomposition * of a real matrix \f$A\f$. The SVD algorithm computes the decomposition * of a real \f$m \times n\f$ matrix \f$\mathbf{A}\f$ so that * \f[ * \mathrm{diag}(\mathbf{U}^T \mathbf{A} \mathbf{V}) = \mathbf{s} * = \sigma_1, \ldots, \sigma_p * \f] * where \f$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0\f$ are the * singular values of \f$\mathbf{A}\f$. * Or put differently: * \f[ * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^T * \f] * where \f$\mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$ * * \note An external LAPACK library is required by this function. */ vec svd(const mat &A); /*! * \ingroup matrixdecomp * \brief Return singular values of a complex matrix \c A using SVD * * This function returns singular values from the SVD * decomposition of a complex matrix \f$A\f$. The SVD algorithm computes * the decomposition of a complex \f$m \times n\f$ matrix \f$\mathbf{A}\f$ * so that * \f[ * \mathrm{diag}(\mathbf{U}^H \mathbf{A} \mathbf{V}) = \mathbf{s} * = \sigma_1, \ldots, \sigma_p * \f] * where \f$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_p \geq 0\f$ * are the singular values of \f$\mathbf{A}\f$. * Or put differently: * \f[ * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H * \f] * where \f$\mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$ * * \note An external LAPACK library is required by this function. */ vec svd(const cmat &A); /*! * \ingroup matrixdecomp * \brief Perform Singular Value Decomposition (SVD) of a real matrix \c A * * This function returns two orthonormal matrices \f$U\f$ and \f$V\f$ * and a vector of singular values \f$s\f$. * The SVD algorithm computes the decomposition of a real \f$m \times n\f$ * matrix \f$\mathbf{A}\f$ so that * \f[ * \mathrm{diag}(\mathbf{U}^T \mathbf{A} \mathbf{V}) = \mathbf{s} * = \sigma_1, \ldots, \sigma_p * \f] * where the elements of \f$\mathbf{s}\f$, \f$\sigma_1 \geq \sigma_2 \geq * \ldots \sigma_p \geq 0\f$ are the singular values of \f$\mathbf{A}\f$. * Or put differently: * \f[ * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^T * \f] * where \f$\mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$ * * \note An external LAPACK library is required by this function. */ bool svd(const mat &A, mat &U, vec &s, mat &V); /*! * \ingroup matrixdecomp * \brief Perform Singular Value Decomposition (SVD) of a complex matrix \c A * * This function returns two orthonormal matrices \f$U\f$ and \f$V\f$ * and a vector of singular values \f$s\f$. * The SVD algorithm computes the decomposition of a complex \f$m \times n\f$ * matrix \f$\mathbf{A}\f$ so that * \f[ * \mathrm{diag}(\mathbf{U}^H \mathbf{A} \mathbf{V}) = \mathbf{s} * = \sigma_1, \ldots, \sigma_p * \f] * where the elements of \f$\mathbf{s}\f$, \f$\sigma_1 \geq \sigma_2 \geq * \ldots \sigma_p \geq 0\f$ are the singular values of \f$\mathbf{A}\f$. * Or put differently: * \f[ * \mathbf{A} = \mathbf{U} \mathbf{S} \mathbf{V}^H * \f] * where \f$\mathrm{diag}(\mathbf{S}) = \mathbf{s} \f$ * * \note An external LAPACK library is required by this function. */ bool svd(const cmat &A, cmat &U, vec &s, cmat &V); } // namespace itpp #endif // #ifndef SVD_H