1 | // This file is part of Eigen, a lightweight C++ template library
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2 | // for linear algebra.
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3 | //
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4 | // Copyright (C) 2009-2010 Benoit Jacob <jacob.benoit.1@gmail.com>
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5 | //
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6 | // This Source Code Form is subject to the terms of the Mozilla
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7 | // Public License v. 2.0. If a copy of the MPL was not distributed
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8 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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9 |
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10 | #ifndef EIGEN_JACOBISVD_H
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11 | #define EIGEN_JACOBISVD_H
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12 |
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13 | namespace Eigen {
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14 |
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15 | namespace internal {
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16 | // forward declaration (needed by ICC)
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17 | // the empty body is required by MSVC
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18 | template<typename MatrixType, int QRPreconditioner,
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19 | bool IsComplex = NumTraits<typename MatrixType::Scalar>::IsComplex>
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20 | struct svd_precondition_2x2_block_to_be_real {};
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21 |
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22 | /*** QR preconditioners (R-SVD)
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23 | ***
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24 | *** Their role is to reduce the problem of computing the SVD to the case of a square matrix.
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25 | *** This approach, known as R-SVD, is an optimization for rectangular-enough matrices, and is a requirement for
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26 | *** JacobiSVD which by itself is only able to work on square matrices.
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27 | ***/
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28 |
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29 | enum { PreconditionIfMoreColsThanRows, PreconditionIfMoreRowsThanCols };
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30 |
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31 | template<typename MatrixType, int QRPreconditioner, int Case>
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32 | struct qr_preconditioner_should_do_anything
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33 | {
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34 | enum { a = MatrixType::RowsAtCompileTime != Dynamic &&
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35 | MatrixType::ColsAtCompileTime != Dynamic &&
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36 | MatrixType::ColsAtCompileTime <= MatrixType::RowsAtCompileTime,
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37 | b = MatrixType::RowsAtCompileTime != Dynamic &&
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38 | MatrixType::ColsAtCompileTime != Dynamic &&
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39 | MatrixType::RowsAtCompileTime <= MatrixType::ColsAtCompileTime,
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40 | ret = !( (QRPreconditioner == NoQRPreconditioner) ||
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41 | (Case == PreconditionIfMoreColsThanRows && bool(a)) ||
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42 | (Case == PreconditionIfMoreRowsThanCols && bool(b)) )
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43 | };
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44 | };
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45 |
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46 | template<typename MatrixType, int QRPreconditioner, int Case,
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47 | bool DoAnything = qr_preconditioner_should_do_anything<MatrixType, QRPreconditioner, Case>::ret
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48 | > struct qr_preconditioner_impl {};
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49 |
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50 | template<typename MatrixType, int QRPreconditioner, int Case>
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51 | class qr_preconditioner_impl<MatrixType, QRPreconditioner, Case, false>
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52 | {
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53 | public:
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54 | typedef typename MatrixType::Index Index;
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55 | void allocate(const JacobiSVD<MatrixType, QRPreconditioner>&) {}
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56 | bool run(JacobiSVD<MatrixType, QRPreconditioner>&, const MatrixType&)
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57 | {
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58 | return false;
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59 | }
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60 | };
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61 |
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62 | /*** preconditioner using FullPivHouseholderQR ***/
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63 |
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64 | template<typename MatrixType>
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65 | class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
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66 | {
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67 | public:
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68 | typedef typename MatrixType::Index Index;
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69 | typedef typename MatrixType::Scalar Scalar;
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70 | enum
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71 | {
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72 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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73 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime
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74 | };
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75 | typedef Matrix<Scalar, 1, RowsAtCompileTime, RowMajor, 1, MaxRowsAtCompileTime> WorkspaceType;
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76 |
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77 | void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd)
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78 | {
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79 | if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
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80 | {
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81 | m_qr.~QRType();
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82 | ::new (&m_qr) QRType(svd.rows(), svd.cols());
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83 | }
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84 | if (svd.m_computeFullU) m_workspace.resize(svd.rows());
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85 | }
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86 |
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87 | bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
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88 | {
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89 | if(matrix.rows() > matrix.cols())
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90 | {
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91 | m_qr.compute(matrix);
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92 | svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
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93 | if(svd.m_computeFullU) m_qr.matrixQ().evalTo(svd.m_matrixU, m_workspace);
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94 | if(svd.computeV()) svd.m_matrixV = m_qr.colsPermutation();
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95 | return true;
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96 | }
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97 | return false;
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98 | }
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99 | private:
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100 | typedef FullPivHouseholderQR<MatrixType> QRType;
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101 | QRType m_qr;
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102 | WorkspaceType m_workspace;
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103 | };
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104 |
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105 | template<typename MatrixType>
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106 | class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
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107 | {
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108 | public:
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109 | typedef typename MatrixType::Index Index;
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110 | typedef typename MatrixType::Scalar Scalar;
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111 | enum
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112 | {
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113 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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114 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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115 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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116 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
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117 | Options = MatrixType::Options
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118 | };
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119 | typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
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120 | TransposeTypeWithSameStorageOrder;
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121 |
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122 | void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd)
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123 | {
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124 | if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
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125 | {
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126 | m_qr.~QRType();
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127 | ::new (&m_qr) QRType(svd.cols(), svd.rows());
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128 | }
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129 | m_adjoint.resize(svd.cols(), svd.rows());
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130 | if (svd.m_computeFullV) m_workspace.resize(svd.cols());
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131 | }
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132 |
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133 | bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
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134 | {
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135 | if(matrix.cols() > matrix.rows())
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136 | {
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137 | m_adjoint = matrix.adjoint();
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138 | m_qr.compute(m_adjoint);
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139 | svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
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140 | if(svd.m_computeFullV) m_qr.matrixQ().evalTo(svd.m_matrixV, m_workspace);
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141 | if(svd.computeU()) svd.m_matrixU = m_qr.colsPermutation();
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142 | return true;
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143 | }
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144 | else return false;
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145 | }
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146 | private:
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147 | typedef FullPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
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148 | QRType m_qr;
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149 | TransposeTypeWithSameStorageOrder m_adjoint;
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150 | typename internal::plain_row_type<MatrixType>::type m_workspace;
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151 | };
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152 |
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153 | /*** preconditioner using ColPivHouseholderQR ***/
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154 |
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155 | template<typename MatrixType>
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156 | class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
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157 | {
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158 | public:
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159 | typedef typename MatrixType::Index Index;
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160 |
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161 | void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd)
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162 | {
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163 | if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
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164 | {
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165 | m_qr.~QRType();
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166 | ::new (&m_qr) QRType(svd.rows(), svd.cols());
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167 | }
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168 | if (svd.m_computeFullU) m_workspace.resize(svd.rows());
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169 | else if (svd.m_computeThinU) m_workspace.resize(svd.cols());
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170 | }
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171 |
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172 | bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
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173 | {
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174 | if(matrix.rows() > matrix.cols())
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175 | {
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176 | m_qr.compute(matrix);
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177 | svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
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178 | if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace);
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179 | else if(svd.m_computeThinU)
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180 | {
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181 | svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
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182 | m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace);
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183 | }
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184 | if(svd.computeV()) svd.m_matrixV = m_qr.colsPermutation();
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185 | return true;
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186 | }
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187 | return false;
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188 | }
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189 |
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190 | private:
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191 | typedef ColPivHouseholderQR<MatrixType> QRType;
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192 | QRType m_qr;
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193 | typename internal::plain_col_type<MatrixType>::type m_workspace;
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194 | };
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195 |
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196 | template<typename MatrixType>
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197 | class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
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198 | {
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199 | public:
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200 | typedef typename MatrixType::Index Index;
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201 | typedef typename MatrixType::Scalar Scalar;
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202 | enum
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203 | {
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204 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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205 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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206 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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207 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
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208 | Options = MatrixType::Options
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209 | };
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210 |
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211 | typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
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212 | TransposeTypeWithSameStorageOrder;
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213 |
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214 | void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd)
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215 | {
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216 | if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
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217 | {
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218 | m_qr.~QRType();
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219 | ::new (&m_qr) QRType(svd.cols(), svd.rows());
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220 | }
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221 | if (svd.m_computeFullV) m_workspace.resize(svd.cols());
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222 | else if (svd.m_computeThinV) m_workspace.resize(svd.rows());
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223 | m_adjoint.resize(svd.cols(), svd.rows());
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224 | }
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225 |
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226 | bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
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227 | {
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228 | if(matrix.cols() > matrix.rows())
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229 | {
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230 | m_adjoint = matrix.adjoint();
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231 | m_qr.compute(m_adjoint);
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232 |
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233 | svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
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234 | if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace);
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235 | else if(svd.m_computeThinV)
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236 | {
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237 | svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
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238 | m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace);
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239 | }
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240 | if(svd.computeU()) svd.m_matrixU = m_qr.colsPermutation();
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241 | return true;
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242 | }
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243 | else return false;
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244 | }
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245 |
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246 | private:
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247 | typedef ColPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
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248 | QRType m_qr;
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249 | TransposeTypeWithSameStorageOrder m_adjoint;
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250 | typename internal::plain_row_type<MatrixType>::type m_workspace;
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251 | };
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252 |
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253 | /*** preconditioner using HouseholderQR ***/
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254 |
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255 | template<typename MatrixType>
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256 | class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
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257 | {
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258 | public:
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259 | typedef typename MatrixType::Index Index;
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260 |
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261 | void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd)
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262 | {
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263 | if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
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264 | {
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265 | m_qr.~QRType();
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266 | ::new (&m_qr) QRType(svd.rows(), svd.cols());
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267 | }
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268 | if (svd.m_computeFullU) m_workspace.resize(svd.rows());
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269 | else if (svd.m_computeThinU) m_workspace.resize(svd.cols());
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270 | }
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271 |
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272 | bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
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273 | {
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274 | if(matrix.rows() > matrix.cols())
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275 | {
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276 | m_qr.compute(matrix);
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277 | svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.cols(),matrix.cols()).template triangularView<Upper>();
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278 | if(svd.m_computeFullU) m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace);
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279 | else if(svd.m_computeThinU)
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280 | {
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281 | svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
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282 | m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace);
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283 | }
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284 | if(svd.computeV()) svd.m_matrixV.setIdentity(matrix.cols(), matrix.cols());
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285 | return true;
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286 | }
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287 | return false;
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288 | }
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289 | private:
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290 | typedef HouseholderQR<MatrixType> QRType;
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291 | QRType m_qr;
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292 | typename internal::plain_col_type<MatrixType>::type m_workspace;
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293 | };
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294 |
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295 | template<typename MatrixType>
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296 | class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
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297 | {
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298 | public:
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299 | typedef typename MatrixType::Index Index;
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300 | typedef typename MatrixType::Scalar Scalar;
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301 | enum
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302 | {
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303 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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304 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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305 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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306 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
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307 | Options = MatrixType::Options
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308 | };
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309 |
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310 | typedef Matrix<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>
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311 | TransposeTypeWithSameStorageOrder;
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312 |
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313 | void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd)
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314 | {
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315 | if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
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316 | {
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317 | m_qr.~QRType();
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318 | ::new (&m_qr) QRType(svd.cols(), svd.rows());
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319 | }
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320 | if (svd.m_computeFullV) m_workspace.resize(svd.cols());
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321 | else if (svd.m_computeThinV) m_workspace.resize(svd.rows());
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322 | m_adjoint.resize(svd.cols(), svd.rows());
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323 | }
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324 |
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325 | bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
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326 | {
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327 | if(matrix.cols() > matrix.rows())
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328 | {
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329 | m_adjoint = matrix.adjoint();
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330 | m_qr.compute(m_adjoint);
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331 |
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332 | svd.m_workMatrix = m_qr.matrixQR().block(0,0,matrix.rows(),matrix.rows()).template triangularView<Upper>().adjoint();
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333 | if(svd.m_computeFullV) m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace);
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334 | else if(svd.m_computeThinV)
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335 | {
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336 | svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
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337 | m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace);
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338 | }
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339 | if(svd.computeU()) svd.m_matrixU.setIdentity(matrix.rows(), matrix.rows());
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340 | return true;
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341 | }
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342 | else return false;
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343 | }
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344 |
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345 | private:
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346 | typedef HouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
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347 | QRType m_qr;
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348 | TransposeTypeWithSameStorageOrder m_adjoint;
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349 | typename internal::plain_row_type<MatrixType>::type m_workspace;
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350 | };
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351 |
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352 | /*** 2x2 SVD implementation
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353 | ***
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354 | *** JacobiSVD consists in performing a series of 2x2 SVD subproblems
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355 | ***/
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356 |
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357 | template<typename MatrixType, int QRPreconditioner>
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358 | struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, false>
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359 | {
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360 | typedef JacobiSVD<MatrixType, QRPreconditioner> SVD;
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361 | typedef typename SVD::Index Index;
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362 | static void run(typename SVD::WorkMatrixType&, SVD&, Index, Index) {}
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363 | };
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364 |
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365 | template<typename MatrixType, int QRPreconditioner>
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366 | struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, true>
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367 | {
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368 | typedef JacobiSVD<MatrixType, QRPreconditioner> SVD;
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369 | typedef typename MatrixType::Scalar Scalar;
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370 | typedef typename MatrixType::RealScalar RealScalar;
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371 | typedef typename SVD::Index Index;
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372 | static void run(typename SVD::WorkMatrixType& work_matrix, SVD& svd, Index p, Index q)
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373 | {
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374 | using std::sqrt;
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375 | Scalar z;
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376 | JacobiRotation<Scalar> rot;
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377 | RealScalar n = sqrt(numext::abs2(work_matrix.coeff(p,p)) + numext::abs2(work_matrix.coeff(q,p)));
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378 | if(n==0)
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379 | {
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380 | z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
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381 | work_matrix.row(p) *= z;
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382 | if(svd.computeU()) svd.m_matrixU.col(p) *= conj(z);
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383 | z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
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384 | work_matrix.row(q) *= z;
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385 | if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
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386 | }
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387 | else
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388 | {
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389 | rot.c() = conj(work_matrix.coeff(p,p)) / n;
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390 | rot.s() = work_matrix.coeff(q,p) / n;
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391 | work_matrix.applyOnTheLeft(p,q,rot);
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392 | if(svd.computeU()) svd.m_matrixU.applyOnTheRight(p,q,rot.adjoint());
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393 | if(work_matrix.coeff(p,q) != Scalar(0))
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394 | {
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395 | Scalar z = abs(work_matrix.coeff(p,q)) / work_matrix.coeff(p,q);
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396 | work_matrix.col(q) *= z;
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397 | if(svd.computeV()) svd.m_matrixV.col(q) *= z;
|
---|
398 | }
|
---|
399 | if(work_matrix.coeff(q,q) != Scalar(0))
|
---|
400 | {
|
---|
401 | z = abs(work_matrix.coeff(q,q)) / work_matrix.coeff(q,q);
|
---|
402 | work_matrix.row(q) *= z;
|
---|
403 | if(svd.computeU()) svd.m_matrixU.col(q) *= conj(z);
|
---|
404 | }
|
---|
405 | }
|
---|
406 | }
|
---|
407 | };
|
---|
408 |
|
---|
409 | template<typename MatrixType, typename RealScalar, typename Index>
|
---|
410 | void real_2x2_jacobi_svd(const MatrixType& matrix, Index p, Index q,
|
---|
411 | JacobiRotation<RealScalar> *j_left,
|
---|
412 | JacobiRotation<RealScalar> *j_right)
|
---|
413 | {
|
---|
414 | using std::sqrt;
|
---|
415 | Matrix<RealScalar,2,2> m;
|
---|
416 | m << numext::real(matrix.coeff(p,p)), numext::real(matrix.coeff(p,q)),
|
---|
417 | numext::real(matrix.coeff(q,p)), numext::real(matrix.coeff(q,q));
|
---|
418 | JacobiRotation<RealScalar> rot1;
|
---|
419 | RealScalar t = m.coeff(0,0) + m.coeff(1,1);
|
---|
420 | RealScalar d = m.coeff(1,0) - m.coeff(0,1);
|
---|
421 | if(t == RealScalar(0))
|
---|
422 | {
|
---|
423 | rot1.c() = RealScalar(0);
|
---|
424 | rot1.s() = d > RealScalar(0) ? RealScalar(1) : RealScalar(-1);
|
---|
425 | }
|
---|
426 | else
|
---|
427 | {
|
---|
428 | RealScalar u = d / t;
|
---|
429 | rot1.c() = RealScalar(1) / sqrt(RealScalar(1) + numext::abs2(u));
|
---|
430 | rot1.s() = rot1.c() * u;
|
---|
431 | }
|
---|
432 | m.applyOnTheLeft(0,1,rot1);
|
---|
433 | j_right->makeJacobi(m,0,1);
|
---|
434 | *j_left = rot1 * j_right->transpose();
|
---|
435 | }
|
---|
436 |
|
---|
437 | } // end namespace internal
|
---|
438 |
|
---|
439 | /** \ingroup SVD_Module
|
---|
440 | *
|
---|
441 | *
|
---|
442 | * \class JacobiSVD
|
---|
443 | *
|
---|
444 | * \brief Two-sided Jacobi SVD decomposition of a rectangular matrix
|
---|
445 | *
|
---|
446 | * \param MatrixType the type of the matrix of which we are computing the SVD decomposition
|
---|
447 | * \param QRPreconditioner this optional parameter allows to specify the type of QR decomposition that will be used internally
|
---|
448 | * for the R-SVD step for non-square matrices. See discussion of possible values below.
|
---|
449 | *
|
---|
450 | * SVD decomposition consists in decomposing any n-by-p matrix \a A as a product
|
---|
451 | * \f[ A = U S V^* \f]
|
---|
452 | * where \a U is a n-by-n unitary, \a V is a p-by-p unitary, and \a S is a n-by-p real positive matrix which is zero outside of its main diagonal;
|
---|
453 | * the diagonal entries of S are known as the \em singular \em values of \a A and the columns of \a U and \a V are known as the left
|
---|
454 | * and right \em singular \em vectors of \a A respectively.
|
---|
455 | *
|
---|
456 | * Singular values are always sorted in decreasing order.
|
---|
457 | *
|
---|
458 | * This JacobiSVD decomposition computes only the singular values by default. If you want \a U or \a V, you need to ask for them explicitly.
|
---|
459 | *
|
---|
460 | * You can ask for only \em thin \a U or \a V to be computed, meaning the following. In case of a rectangular n-by-p matrix, letting \a m be the
|
---|
461 | * smaller value among \a n and \a p, there are only \a m singular vectors; the remaining columns of \a U and \a V do not correspond to actual
|
---|
462 | * singular vectors. Asking for \em thin \a U or \a V means asking for only their \a m first columns to be formed. So \a U is then a n-by-m matrix,
|
---|
463 | * and \a V is then a p-by-m matrix. Notice that thin \a U and \a V are all you need for (least squares) solving.
|
---|
464 | *
|
---|
465 | * Here's an example demonstrating basic usage:
|
---|
466 | * \include JacobiSVD_basic.cpp
|
---|
467 | * Output: \verbinclude JacobiSVD_basic.out
|
---|
468 | *
|
---|
469 | * This JacobiSVD class is a two-sided Jacobi R-SVD decomposition, ensuring optimal reliability and accuracy. The downside is that it's slower than
|
---|
470 | * bidiagonalizing SVD algorithms for large square matrices; however its complexity is still \f$ O(n^2p) \f$ where \a n is the smaller dimension and
|
---|
471 | * \a p is the greater dimension, meaning that it is still of the same order of complexity as the faster bidiagonalizing R-SVD algorithms.
|
---|
472 | * In particular, like any R-SVD, it takes advantage of non-squareness in that its complexity is only linear in the greater dimension.
|
---|
473 | *
|
---|
474 | * If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is guaranteed to
|
---|
475 | * terminate in finite (and reasonable) time.
|
---|
476 | *
|
---|
477 | * The possible values for QRPreconditioner are:
|
---|
478 | * \li ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR.
|
---|
479 | * \li FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR.
|
---|
480 | * Contrary to other QRs, it doesn't allow computing thin unitaries.
|
---|
481 | * \li HouseholderQRPreconditioner is the fastest, and less safe and accurate than the pivoting variants. It uses non-pivoting QR.
|
---|
482 | * This is very similar in safety and accuracy to the bidiagonalization process used by bidiagonalizing SVD algorithms (since bidiagonalization
|
---|
483 | * is inherently non-pivoting). However the resulting SVD is still more reliable than bidiagonalizing SVDs because the Jacobi-based iterarive
|
---|
484 | * process is more reliable than the optimized bidiagonal SVD iterations.
|
---|
485 | * \li NoQRPreconditioner allows not to use a QR preconditioner at all. This is useful if you know that you will only be computing
|
---|
486 | * JacobiSVD decompositions of square matrices. Non-square matrices require a QR preconditioner. Using this option will result in
|
---|
487 | * faster compilation and smaller executable code. It won't significantly speed up computation, since JacobiSVD is always checking
|
---|
488 | * if QR preconditioning is needed before applying it anyway.
|
---|
489 | *
|
---|
490 | * \sa MatrixBase::jacobiSvd()
|
---|
491 | */
|
---|
492 | template<typename _MatrixType, int QRPreconditioner>
|
---|
493 | class JacobiSVD : public SVDBase<_MatrixType>
|
---|
494 | {
|
---|
495 | public:
|
---|
496 |
|
---|
497 | typedef _MatrixType MatrixType;
|
---|
498 | typedef typename MatrixType::Scalar Scalar;
|
---|
499 | typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
|
---|
500 | typedef typename MatrixType::Index Index;
|
---|
501 | enum {
|
---|
502 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
|
---|
503 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
|
---|
504 | DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime,ColsAtCompileTime),
|
---|
505 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
|
---|
506 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
|
---|
507 | MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime,MaxColsAtCompileTime),
|
---|
508 | MatrixOptions = MatrixType::Options
|
---|
509 | };
|
---|
510 |
|
---|
511 | typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime,
|
---|
512 | MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime>
|
---|
513 | MatrixUType;
|
---|
514 | typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime,
|
---|
515 | MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime>
|
---|
516 | MatrixVType;
|
---|
517 | typedef typename internal::plain_diag_type<MatrixType, RealScalar>::type SingularValuesType;
|
---|
518 | typedef typename internal::plain_row_type<MatrixType>::type RowType;
|
---|
519 | typedef typename internal::plain_col_type<MatrixType>::type ColType;
|
---|
520 | typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime,
|
---|
521 | MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime>
|
---|
522 | WorkMatrixType;
|
---|
523 |
|
---|
524 | /** \brief Default Constructor.
|
---|
525 | *
|
---|
526 | * The default constructor is useful in cases in which the user intends to
|
---|
527 | * perform decompositions via JacobiSVD::compute(const MatrixType&).
|
---|
528 | */
|
---|
529 | JacobiSVD()
|
---|
530 | : SVDBase<_MatrixType>::SVDBase()
|
---|
531 | {}
|
---|
532 |
|
---|
533 |
|
---|
534 | /** \brief Default Constructor with memory preallocation
|
---|
535 | *
|
---|
536 | * Like the default constructor but with preallocation of the internal data
|
---|
537 | * according to the specified problem size.
|
---|
538 | * \sa JacobiSVD()
|
---|
539 | */
|
---|
540 | JacobiSVD(Index rows, Index cols, unsigned int computationOptions = 0)
|
---|
541 | : SVDBase<_MatrixType>::SVDBase()
|
---|
542 | {
|
---|
543 | allocate(rows, cols, computationOptions);
|
---|
544 | }
|
---|
545 |
|
---|
546 | /** \brief Constructor performing the decomposition of given matrix.
|
---|
547 | *
|
---|
548 | * \param matrix the matrix to decompose
|
---|
549 | * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
|
---|
550 | * By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU,
|
---|
551 | * #ComputeFullV, #ComputeThinV.
|
---|
552 | *
|
---|
553 | * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
|
---|
554 | * available with the (non-default) FullPivHouseholderQR preconditioner.
|
---|
555 | */
|
---|
556 | JacobiSVD(const MatrixType& matrix, unsigned int computationOptions = 0)
|
---|
557 | : SVDBase<_MatrixType>::SVDBase()
|
---|
558 | {
|
---|
559 | compute(matrix, computationOptions);
|
---|
560 | }
|
---|
561 |
|
---|
562 | /** \brief Method performing the decomposition of given matrix using custom options.
|
---|
563 | *
|
---|
564 | * \param matrix the matrix to decompose
|
---|
565 | * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
|
---|
566 | * By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU,
|
---|
567 | * #ComputeFullV, #ComputeThinV.
|
---|
568 | *
|
---|
569 | * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
|
---|
570 | * available with the (non-default) FullPivHouseholderQR preconditioner.
|
---|
571 | */
|
---|
572 | SVDBase<MatrixType>& compute(const MatrixType& matrix, unsigned int computationOptions);
|
---|
573 |
|
---|
574 | /** \brief Method performing the decomposition of given matrix using current options.
|
---|
575 | *
|
---|
576 | * \param matrix the matrix to decompose
|
---|
577 | *
|
---|
578 | * This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).
|
---|
579 | */
|
---|
580 | SVDBase<MatrixType>& compute(const MatrixType& matrix)
|
---|
581 | {
|
---|
582 | return compute(matrix, this->m_computationOptions);
|
---|
583 | }
|
---|
584 |
|
---|
585 | /** \returns a (least squares) solution of \f$ A x = b \f$ using the current SVD decomposition of A.
|
---|
586 | *
|
---|
587 | * \param b the right-hand-side of the equation to solve.
|
---|
588 | *
|
---|
589 | * \note Solving requires both U and V to be computed. Thin U and V are enough, there is no need for full U or V.
|
---|
590 | *
|
---|
591 | * \note SVD solving is implicitly least-squares. Thus, this method serves both purposes of exact solving and least-squares solving.
|
---|
592 | * In other words, the returned solution is guaranteed to minimize the Euclidean norm \f$ \Vert A x - b \Vert \f$.
|
---|
593 | */
|
---|
594 | template<typename Rhs>
|
---|
595 | inline const internal::solve_retval<JacobiSVD, Rhs>
|
---|
596 | solve(const MatrixBase<Rhs>& b) const
|
---|
597 | {
|
---|
598 | eigen_assert(this->m_isInitialized && "JacobiSVD is not initialized.");
|
---|
599 | eigen_assert(SVDBase<MatrixType>::computeU() && SVDBase<MatrixType>::computeV() && "JacobiSVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice).");
|
---|
600 | return internal::solve_retval<JacobiSVD, Rhs>(*this, b.derived());
|
---|
601 | }
|
---|
602 |
|
---|
603 |
|
---|
604 |
|
---|
605 | private:
|
---|
606 | void allocate(Index rows, Index cols, unsigned int computationOptions);
|
---|
607 |
|
---|
608 | protected:
|
---|
609 | WorkMatrixType m_workMatrix;
|
---|
610 |
|
---|
611 | template<typename __MatrixType, int _QRPreconditioner, bool _IsComplex>
|
---|
612 | friend struct internal::svd_precondition_2x2_block_to_be_real;
|
---|
613 | template<typename __MatrixType, int _QRPreconditioner, int _Case, bool _DoAnything>
|
---|
614 | friend struct internal::qr_preconditioner_impl;
|
---|
615 |
|
---|
616 | internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> m_qr_precond_morecols;
|
---|
617 | internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> m_qr_precond_morerows;
|
---|
618 | };
|
---|
619 |
|
---|
620 | template<typename MatrixType, int QRPreconditioner>
|
---|
621 | void JacobiSVD<MatrixType, QRPreconditioner>::allocate(Index rows, Index cols, unsigned int computationOptions)
|
---|
622 | {
|
---|
623 | if (SVDBase<MatrixType>::allocate(rows, cols, computationOptions)) return;
|
---|
624 |
|
---|
625 | if (QRPreconditioner == FullPivHouseholderQRPreconditioner)
|
---|
626 | {
|
---|
627 | eigen_assert(!(this->m_computeThinU || this->m_computeThinV) &&
|
---|
628 | "JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. "
|
---|
629 | "Use the ColPivHouseholderQR preconditioner instead.");
|
---|
630 | }
|
---|
631 |
|
---|
632 | m_workMatrix.resize(this->m_diagSize, this->m_diagSize);
|
---|
633 |
|
---|
634 | if(this->m_cols>this->m_rows) m_qr_precond_morecols.allocate(*this);
|
---|
635 | if(this->m_rows>this->m_cols) m_qr_precond_morerows.allocate(*this);
|
---|
636 | }
|
---|
637 |
|
---|
638 | template<typename MatrixType, int QRPreconditioner>
|
---|
639 | SVDBase<MatrixType>&
|
---|
640 | JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsigned int computationOptions)
|
---|
641 | {
|
---|
642 | using std::abs;
|
---|
643 | allocate(matrix.rows(), matrix.cols(), computationOptions);
|
---|
644 |
|
---|
645 | // currently we stop when we reach precision 2*epsilon as the last bit of precision can require an unreasonable number of iterations,
|
---|
646 | // only worsening the precision of U and V as we accumulate more rotations
|
---|
647 | const RealScalar precision = RealScalar(2) * NumTraits<Scalar>::epsilon();
|
---|
648 |
|
---|
649 | // limit for very small denormal numbers to be considered zero in order to avoid infinite loops (see bug 286)
|
---|
650 | const RealScalar considerAsZero = RealScalar(2) * std::numeric_limits<RealScalar>::denorm_min();
|
---|
651 |
|
---|
652 | /*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */
|
---|
653 |
|
---|
654 | if(!m_qr_precond_morecols.run(*this, matrix) && !m_qr_precond_morerows.run(*this, matrix))
|
---|
655 | {
|
---|
656 | m_workMatrix = matrix.block(0,0,this->m_diagSize,this->m_diagSize);
|
---|
657 | if(this->m_computeFullU) this->m_matrixU.setIdentity(this->m_rows,this->m_rows);
|
---|
658 | if(this->m_computeThinU) this->m_matrixU.setIdentity(this->m_rows,this->m_diagSize);
|
---|
659 | if(this->m_computeFullV) this->m_matrixV.setIdentity(this->m_cols,this->m_cols);
|
---|
660 | if(this->m_computeThinV) this->m_matrixV.setIdentity(this->m_cols, this->m_diagSize);
|
---|
661 | }
|
---|
662 |
|
---|
663 | /*** step 2. The main Jacobi SVD iteration. ***/
|
---|
664 |
|
---|
665 | bool finished = false;
|
---|
666 | while(!finished)
|
---|
667 | {
|
---|
668 | finished = true;
|
---|
669 |
|
---|
670 | // do a sweep: for all index pairs (p,q), perform SVD of the corresponding 2x2 sub-matrix
|
---|
671 |
|
---|
672 | for(Index p = 1; p < this->m_diagSize; ++p)
|
---|
673 | {
|
---|
674 | for(Index q = 0; q < p; ++q)
|
---|
675 | {
|
---|
676 | // if this 2x2 sub-matrix is not diagonal already...
|
---|
677 | // notice that this comparison will evaluate to false if any NaN is involved, ensuring that NaN's don't
|
---|
678 | // keep us iterating forever. Similarly, small denormal numbers are considered zero.
|
---|
679 | using std::max;
|
---|
680 | RealScalar threshold = (max)(considerAsZero, precision * (max)(abs(m_workMatrix.coeff(p,p)),
|
---|
681 | abs(m_workMatrix.coeff(q,q))));
|
---|
682 | if((max)(abs(m_workMatrix.coeff(p,q)),abs(m_workMatrix.coeff(q,p))) > threshold)
|
---|
683 | {
|
---|
684 | finished = false;
|
---|
685 |
|
---|
686 | // perform SVD decomposition of 2x2 sub-matrix corresponding to indices p,q to make it diagonal
|
---|
687 | internal::svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner>::run(m_workMatrix, *this, p, q);
|
---|
688 | JacobiRotation<RealScalar> j_left, j_right;
|
---|
689 | internal::real_2x2_jacobi_svd(m_workMatrix, p, q, &j_left, &j_right);
|
---|
690 |
|
---|
691 | // accumulate resulting Jacobi rotations
|
---|
692 | m_workMatrix.applyOnTheLeft(p,q,j_left);
|
---|
693 | if(SVDBase<MatrixType>::computeU()) this->m_matrixU.applyOnTheRight(p,q,j_left.transpose());
|
---|
694 |
|
---|
695 | m_workMatrix.applyOnTheRight(p,q,j_right);
|
---|
696 | if(SVDBase<MatrixType>::computeV()) this->m_matrixV.applyOnTheRight(p,q,j_right);
|
---|
697 | }
|
---|
698 | }
|
---|
699 | }
|
---|
700 | }
|
---|
701 |
|
---|
702 | /*** step 3. The work matrix is now diagonal, so ensure it's positive so its diagonal entries are the singular values ***/
|
---|
703 |
|
---|
704 | for(Index i = 0; i < this->m_diagSize; ++i)
|
---|
705 | {
|
---|
706 | RealScalar a = abs(m_workMatrix.coeff(i,i));
|
---|
707 | this->m_singularValues.coeffRef(i) = a;
|
---|
708 | if(SVDBase<MatrixType>::computeU() && (a!=RealScalar(0))) this->m_matrixU.col(i) *= this->m_workMatrix.coeff(i,i)/a;
|
---|
709 | }
|
---|
710 |
|
---|
711 | /*** step 4. Sort singular values in descending order and compute the number of nonzero singular values ***/
|
---|
712 |
|
---|
713 | this->m_nonzeroSingularValues = this->m_diagSize;
|
---|
714 | for(Index i = 0; i < this->m_diagSize; i++)
|
---|
715 | {
|
---|
716 | Index pos;
|
---|
717 | RealScalar maxRemainingSingularValue = this->m_singularValues.tail(this->m_diagSize-i).maxCoeff(&pos);
|
---|
718 | if(maxRemainingSingularValue == RealScalar(0))
|
---|
719 | {
|
---|
720 | this->m_nonzeroSingularValues = i;
|
---|
721 | break;
|
---|
722 | }
|
---|
723 | if(pos)
|
---|
724 | {
|
---|
725 | pos += i;
|
---|
726 | std::swap(this->m_singularValues.coeffRef(i), this->m_singularValues.coeffRef(pos));
|
---|
727 | if(SVDBase<MatrixType>::computeU()) this->m_matrixU.col(pos).swap(this->m_matrixU.col(i));
|
---|
728 | if(SVDBase<MatrixType>::computeV()) this->m_matrixV.col(pos).swap(this->m_matrixV.col(i));
|
---|
729 | }
|
---|
730 | }
|
---|
731 |
|
---|
732 | this->m_isInitialized = true;
|
---|
733 | return *this;
|
---|
734 | }
|
---|
735 |
|
---|
736 | namespace internal {
|
---|
737 | template<typename _MatrixType, int QRPreconditioner, typename Rhs>
|
---|
738 | struct solve_retval<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs>
|
---|
739 | : solve_retval_base<JacobiSVD<_MatrixType, QRPreconditioner>, Rhs>
|
---|
740 | {
|
---|
741 | typedef JacobiSVD<_MatrixType, QRPreconditioner> JacobiSVDType;
|
---|
742 | EIGEN_MAKE_SOLVE_HELPERS(JacobiSVDType,Rhs)
|
---|
743 |
|
---|
744 | template<typename Dest> void evalTo(Dest& dst) const
|
---|
745 | {
|
---|
746 | eigen_assert(rhs().rows() == dec().rows());
|
---|
747 |
|
---|
748 | // A = U S V^*
|
---|
749 | // So A^{-1} = V S^{-1} U^*
|
---|
750 |
|
---|
751 | Index diagSize = (std::min)(dec().rows(), dec().cols());
|
---|
752 | typename JacobiSVDType::SingularValuesType invertedSingVals(diagSize);
|
---|
753 |
|
---|
754 | Index nonzeroSingVals = dec().nonzeroSingularValues();
|
---|
755 | invertedSingVals.head(nonzeroSingVals) = dec().singularValues().head(nonzeroSingVals).array().inverse();
|
---|
756 | invertedSingVals.tail(diagSize - nonzeroSingVals).setZero();
|
---|
757 |
|
---|
758 | dst = dec().matrixV().leftCols(diagSize)
|
---|
759 | * invertedSingVals.asDiagonal()
|
---|
760 | * dec().matrixU().leftCols(diagSize).adjoint()
|
---|
761 | * rhs();
|
---|
762 | }
|
---|
763 | };
|
---|
764 | } // end namespace internal
|
---|
765 |
|
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766 | /** \svd_module
|
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767 | *
|
---|
768 | * \return the singular value decomposition of \c *this computed by two-sided
|
---|
769 | * Jacobi transformations.
|
---|
770 | *
|
---|
771 | * \sa class JacobiSVD
|
---|
772 | */
|
---|
773 | template<typename Derived>
|
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774 | JacobiSVD<typename MatrixBase<Derived>::PlainObject>
|
---|
775 | MatrixBase<Derived>::jacobiSvd(unsigned int computationOptions) const
|
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776 | {
|
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777 | return JacobiSVD<PlainObject>(*this, computationOptions);
|
---|
778 | }
|
---|
779 |
|
---|
780 | } // end namespace Eigen
|
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781 |
|
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782 | #endif // EIGEN_JACOBISVD_H
|
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