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 | // We used the "A Divide-And-Conquer Algorithm for the Bidiagonal SVD"
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5 | // research report written by Ming Gu and Stanley C.Eisenstat
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6 | // The code variable names correspond to the names they used in their
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7 | // report
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8 | //
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9 | // Copyright (C) 2013 Gauthier Brun <brun.gauthier@gmail.com>
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10 | // Copyright (C) 2013 Nicolas Carre <nicolas.carre@ensimag.fr>
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11 | // Copyright (C) 2013 Jean Ceccato <jean.ceccato@ensimag.fr>
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12 | // Copyright (C) 2013 Pierre Zoppitelli <pierre.zoppitelli@ensimag.fr>
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13 | //
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14 | // Source Code Form is subject to the terms of the Mozilla
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15 | // Public License v. 2.0. If a copy of the MPL was not distributed
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16 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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17 |
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18 | #ifndef EIGEN_BDCSVD_H
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19 | #define EIGEN_BDCSVD_H
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20 |
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21 | #define EPSILON 0.0000000000000001
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22 |
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23 | #define ALGOSWAP 32
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24 |
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25 | namespace Eigen {
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26 | /** \ingroup SVD_Module
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27 | *
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28 | *
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29 | * \class BDCSVD
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30 | *
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31 | * \brief class Bidiagonal Divide and Conquer SVD
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32 | *
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33 | * \param MatrixType the type of the matrix of which we are computing the SVD decomposition
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34 | * We plan to have a very similar interface to JacobiSVD on this class.
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35 | * It should be used to speed up the calcul of SVD for big matrices.
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36 | */
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37 | template<typename _MatrixType>
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38 | class BDCSVD : public SVDBase<_MatrixType>
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39 | {
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40 | typedef SVDBase<_MatrixType> Base;
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41 |
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42 | public:
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43 | using Base::rows;
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44 | using Base::cols;
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45 |
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46 | typedef _MatrixType MatrixType;
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47 | typedef typename MatrixType::Scalar Scalar;
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48 | typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
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49 | typedef typename MatrixType::Index Index;
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50 | enum {
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51 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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52 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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53 | DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime, ColsAtCompileTime),
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54 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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55 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
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56 | MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime, MaxColsAtCompileTime),
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57 | MatrixOptions = MatrixType::Options
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58 | };
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59 |
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60 | typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime,
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61 | MatrixOptions, MaxRowsAtCompileTime, MaxRowsAtCompileTime>
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62 | MatrixUType;
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63 | typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime,
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64 | MatrixOptions, MaxColsAtCompileTime, MaxColsAtCompileTime>
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65 | MatrixVType;
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66 | typedef typename internal::plain_diag_type<MatrixType, RealScalar>::type SingularValuesType;
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67 | typedef typename internal::plain_row_type<MatrixType>::type RowType;
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68 | typedef typename internal::plain_col_type<MatrixType>::type ColType;
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69 | typedef Matrix<Scalar, Dynamic, Dynamic> MatrixX;
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70 | typedef Matrix<RealScalar, Dynamic, Dynamic> MatrixXr;
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71 | typedef Matrix<RealScalar, Dynamic, 1> VectorType;
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72 |
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73 | /** \brief Default Constructor.
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74 | *
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75 | * The default constructor is useful in cases in which the user intends to
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76 | * perform decompositions via BDCSVD::compute(const MatrixType&).
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77 | */
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78 | BDCSVD()
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79 | : SVDBase<_MatrixType>::SVDBase(),
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80 | algoswap(ALGOSWAP)
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81 | {}
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82 |
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83 |
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84 | /** \brief Default Constructor with memory preallocation
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85 | *
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86 | * Like the default constructor but with preallocation of the internal data
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87 | * according to the specified problem size.
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88 | * \sa BDCSVD()
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89 | */
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90 | BDCSVD(Index rows, Index cols, unsigned int computationOptions = 0)
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91 | : SVDBase<_MatrixType>::SVDBase(),
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92 | algoswap(ALGOSWAP)
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93 | {
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94 | allocate(rows, cols, computationOptions);
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95 | }
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96 |
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97 | /** \brief Constructor performing the decomposition of given matrix.
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98 | *
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99 | * \param matrix the matrix to decompose
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100 | * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
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101 | * By default, none is computed. This is a bit - field, the possible bits are #ComputeFullU, #ComputeThinU,
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102 | * #ComputeFullV, #ComputeThinV.
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103 | *
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104 | * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
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105 | * available with the (non - default) FullPivHouseholderQR preconditioner.
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106 | */
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107 | BDCSVD(const MatrixType& matrix, unsigned int computationOptions = 0)
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108 | : SVDBase<_MatrixType>::SVDBase(),
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109 | algoswap(ALGOSWAP)
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110 | {
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111 | compute(matrix, computationOptions);
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112 | }
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113 |
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114 | ~BDCSVD()
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115 | {
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116 | }
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117 | /** \brief Method performing the decomposition of given matrix using custom options.
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118 | *
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119 | * \param matrix the matrix to decompose
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120 | * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
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121 | * By default, none is computed. This is a bit - field, the possible bits are #ComputeFullU, #ComputeThinU,
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122 | * #ComputeFullV, #ComputeThinV.
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123 | *
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124 | * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
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125 | * available with the (non - default) FullPivHouseholderQR preconditioner.
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126 | */
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127 | SVDBase<MatrixType>& compute(const MatrixType& matrix, unsigned int computationOptions);
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128 |
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129 | /** \brief Method performing the decomposition of given matrix using current options.
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130 | *
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131 | * \param matrix the matrix to decompose
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132 | *
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133 | * This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).
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134 | */
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135 | SVDBase<MatrixType>& compute(const MatrixType& matrix)
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136 | {
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137 | return compute(matrix, this->m_computationOptions);
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138 | }
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139 |
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140 | void setSwitchSize(int s)
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141 | {
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142 | eigen_assert(s>3 && "BDCSVD the size of the algo switch has to be greater than 4");
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143 | algoswap = s;
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144 | }
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145 |
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146 |
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147 | /** \returns a (least squares) solution of \f$ A x = b \f$ using the current SVD decomposition of A.
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148 | *
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149 | * \param b the right - hand - side of the equation to solve.
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150 | *
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151 | * \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.
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152 | *
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153 | * \note SVD solving is implicitly least - squares. Thus, this method serves both purposes of exact solving and least - squares solving.
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154 | * In other words, the returned solution is guaranteed to minimize the Euclidean norm \f$ \Vert A x - b \Vert \f$.
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155 | */
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156 | template<typename Rhs>
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157 | inline const internal::solve_retval<BDCSVD, Rhs>
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158 | solve(const MatrixBase<Rhs>& b) const
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159 | {
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160 | eigen_assert(this->m_isInitialized && "BDCSVD is not initialized.");
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161 | eigen_assert(SVDBase<_MatrixType>::computeU() && SVDBase<_MatrixType>::computeV() &&
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162 | "BDCSVD::solve() requires both unitaries U and V to be computed (thin unitaries suffice).");
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163 | return internal::solve_retval<BDCSVD, Rhs>(*this, b.derived());
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164 | }
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165 |
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166 |
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167 | const MatrixUType& matrixU() const
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168 | {
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169 | eigen_assert(this->m_isInitialized && "SVD is not initialized.");
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170 | if (isTranspose){
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171 | eigen_assert(this->computeV() && "This SVD decomposition didn't compute U. Did you ask for it?");
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172 | return this->m_matrixV;
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173 | }
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174 | else
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175 | {
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176 | eigen_assert(this->computeU() && "This SVD decomposition didn't compute U. Did you ask for it?");
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177 | return this->m_matrixU;
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178 | }
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179 |
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180 | }
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181 |
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182 |
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183 | const MatrixVType& matrixV() const
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184 | {
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185 | eigen_assert(this->m_isInitialized && "SVD is not initialized.");
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186 | if (isTranspose){
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187 | eigen_assert(this->computeU() && "This SVD decomposition didn't compute V. Did you ask for it?");
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188 | return this->m_matrixU;
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189 | }
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190 | else
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191 | {
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192 | eigen_assert(this->computeV() && "This SVD decomposition didn't compute V. Did you ask for it?");
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193 | return this->m_matrixV;
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194 | }
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195 | }
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196 |
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197 | private:
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198 | void allocate(Index rows, Index cols, unsigned int computationOptions);
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199 | void divide (Index firstCol, Index lastCol, Index firstRowW,
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200 | Index firstColW, Index shift);
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201 | void deflation43(Index firstCol, Index shift, Index i, Index size);
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202 | void deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size);
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203 | void deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift);
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204 | void copyUV(MatrixXr naiveU, MatrixXr naiveV, MatrixX householderU, MatrixX houseHolderV);
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205 |
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206 | protected:
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207 | MatrixXr m_naiveU, m_naiveV;
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208 | MatrixXr m_computed;
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209 | Index nRec;
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210 | int algoswap;
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211 | bool isTranspose, compU, compV;
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212 |
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213 | }; //end class BDCSVD
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214 |
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215 |
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216 | // Methode to allocate ans initialize matrix and attributs
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217 | template<typename MatrixType>
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218 | void BDCSVD<MatrixType>::allocate(Index rows, Index cols, unsigned int computationOptions)
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219 | {
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220 | isTranspose = (cols > rows);
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221 | if (SVDBase<MatrixType>::allocate(rows, cols, computationOptions)) return;
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222 | m_computed = MatrixXr::Zero(this->m_diagSize + 1, this->m_diagSize );
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223 | if (isTranspose){
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224 | compU = this->computeU();
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225 | compV = this->computeV();
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226 | }
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227 | else
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228 | {
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229 | compV = this->computeU();
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230 | compU = this->computeV();
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231 | }
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232 | if (compU) m_naiveU = MatrixXr::Zero(this->m_diagSize + 1, this->m_diagSize + 1 );
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233 | else m_naiveU = MatrixXr::Zero(2, this->m_diagSize + 1 );
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234 |
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235 | if (compV) m_naiveV = MatrixXr::Zero(this->m_diagSize, this->m_diagSize);
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236 |
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237 |
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238 | //should be changed for a cleaner implementation
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239 | if (isTranspose){
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240 | bool aux;
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241 | if (this->computeU()||this->computeV()){
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242 | aux = this->m_computeFullU;
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243 | this->m_computeFullU = this->m_computeFullV;
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244 | this->m_computeFullV = aux;
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245 | aux = this->m_computeThinU;
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246 | this->m_computeThinU = this->m_computeThinV;
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247 | this->m_computeThinV = aux;
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248 | }
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249 | }
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250 | }// end allocate
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251 |
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252 | // Methode which compute the BDCSVD for the int
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253 | template<>
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254 | SVDBase<Matrix<int, Dynamic, Dynamic> >&
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255 | BDCSVD<Matrix<int, Dynamic, Dynamic> >::compute(const MatrixType& matrix, unsigned int computationOptions) {
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256 | allocate(matrix.rows(), matrix.cols(), computationOptions);
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257 | this->m_nonzeroSingularValues = 0;
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258 | m_computed = Matrix<int, Dynamic, Dynamic>::Zero(rows(), cols());
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259 | for (int i=0; i<this->m_diagSize; i++) {
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260 | this->m_singularValues.coeffRef(i) = 0;
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261 | }
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262 | if (this->m_computeFullU) this->m_matrixU = Matrix<int, Dynamic, Dynamic>::Zero(rows(), rows());
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263 | if (this->m_computeFullV) this->m_matrixV = Matrix<int, Dynamic, Dynamic>::Zero(cols(), cols());
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264 | this->m_isInitialized = true;
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265 | return *this;
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266 | }
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267 |
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268 |
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269 | // Methode which compute the BDCSVD
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270 | template<typename MatrixType>
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271 | SVDBase<MatrixType>&
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272 | BDCSVD<MatrixType>::compute(const MatrixType& matrix, unsigned int computationOptions)
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273 | {
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274 | allocate(matrix.rows(), matrix.cols(), computationOptions);
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275 | using std::abs;
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276 |
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277 | //**** step 1 Bidiagonalization isTranspose = (matrix.cols()>matrix.rows()) ;
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278 | MatrixType copy;
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279 | if (isTranspose) copy = matrix.adjoint();
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280 | else copy = matrix;
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281 |
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282 | internal::UpperBidiagonalization<MatrixX > bid(copy);
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283 |
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284 | //**** step 2 Divide
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285 | // this is ugly and has to be redone (care of complex cast)
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286 | MatrixXr temp;
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287 | temp = bid.bidiagonal().toDenseMatrix().transpose();
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288 | m_computed.setZero();
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289 | for (int i=0; i<this->m_diagSize - 1; i++) {
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290 | m_computed(i, i) = temp(i, i);
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291 | m_computed(i + 1, i) = temp(i + 1, i);
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292 | }
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293 | m_computed(this->m_diagSize - 1, this->m_diagSize - 1) = temp(this->m_diagSize - 1, this->m_diagSize - 1);
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294 | divide(0, this->m_diagSize - 1, 0, 0, 0);
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295 |
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296 | //**** step 3 copy
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297 | for (int i=0; i<this->m_diagSize; i++) {
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298 | RealScalar a = abs(m_computed.coeff(i, i));
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299 | this->m_singularValues.coeffRef(i) = a;
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300 | if (a == 0){
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301 | this->m_nonzeroSingularValues = i;
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302 | break;
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303 | }
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304 | else if (i == this->m_diagSize - 1)
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305 | {
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306 | this->m_nonzeroSingularValues = i + 1;
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307 | break;
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308 | }
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309 | }
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310 | copyUV(m_naiveV, m_naiveU, bid.householderU(), bid.householderV());
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311 | this->m_isInitialized = true;
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312 | return *this;
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313 | }// end compute
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314 |
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315 |
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316 | template<typename MatrixType>
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317 | void BDCSVD<MatrixType>::copyUV(MatrixXr naiveU, MatrixXr naiveV, MatrixX householderU, MatrixX householderV){
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318 | if (this->computeU()){
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319 | MatrixX temp = MatrixX::Zero(naiveU.rows(), naiveU.cols());
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320 | temp.real() = naiveU;
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321 | if (this->m_computeThinU){
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322 | this->m_matrixU = MatrixX::Identity(householderU.cols(), this->m_nonzeroSingularValues );
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323 | this->m_matrixU.block(0, 0, this->m_diagSize, this->m_nonzeroSingularValues) =
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324 | temp.block(0, 0, this->m_diagSize, this->m_nonzeroSingularValues);
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325 | this->m_matrixU = householderU * this->m_matrixU ;
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326 | }
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327 | else
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328 | {
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329 | this->m_matrixU = MatrixX::Identity(householderU.cols(), householderU.cols());
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330 | this->m_matrixU.block(0, 0, this->m_diagSize, this->m_diagSize) = temp.block(0, 0, this->m_diagSize, this->m_diagSize);
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331 | this->m_matrixU = householderU * this->m_matrixU ;
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332 | }
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333 | }
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334 | if (this->computeV()){
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335 | MatrixX temp = MatrixX::Zero(naiveV.rows(), naiveV.cols());
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336 | temp.real() = naiveV;
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337 | if (this->m_computeThinV){
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338 | this->m_matrixV = MatrixX::Identity(householderV.cols(),this->m_nonzeroSingularValues );
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339 | this->m_matrixV.block(0, 0, this->m_nonzeroSingularValues, this->m_nonzeroSingularValues) =
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340 | temp.block(0, 0, this->m_nonzeroSingularValues, this->m_nonzeroSingularValues);
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341 | this->m_matrixV = householderV * this->m_matrixV ;
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342 | }
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343 | else
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344 | {
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345 | this->m_matrixV = MatrixX::Identity(householderV.cols(), householderV.cols());
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346 | this->m_matrixV.block(0, 0, this->m_diagSize, this->m_diagSize) = temp.block(0, 0, this->m_diagSize, this->m_diagSize);
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347 | this->m_matrixV = householderV * this->m_matrixV;
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348 | }
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349 | }
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350 | }
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351 |
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352 | // The divide algorithm is done "in place", we are always working on subsets of the same matrix. The divide methods takes as argument the
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353 | // place of the submatrix we are currently working on.
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354 |
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355 | //@param firstCol : The Index of the first column of the submatrix of m_computed and for m_naiveU;
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356 | //@param lastCol : The Index of the last column of the submatrix of m_computed and for m_naiveU;
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357 | // lastCol + 1 - firstCol is the size of the submatrix.
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358 | //@param firstRowW : The Index of the first row of the matrix W that we are to change. (see the reference paper section 1 for more information on W)
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359 | //@param firstRowW : Same as firstRowW with the column.
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360 | //@param shift : Each time one takes the left submatrix, one must add 1 to the shift. Why? Because! We actually want the last column of the U submatrix
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361 | // to become the first column (*coeff) and to shift all the other columns to the right. There are more details on the reference paper.
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362 | template<typename MatrixType>
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363 | void BDCSVD<MatrixType>::divide (Index firstCol, Index lastCol, Index firstRowW,
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364 | Index firstColW, Index shift)
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365 | {
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366 | // requires nbRows = nbCols + 1;
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367 | using std::pow;
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368 | using std::sqrt;
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369 | using std::abs;
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370 | const Index n = lastCol - firstCol + 1;
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371 | const Index k = n/2;
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372 | RealScalar alphaK;
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373 | RealScalar betaK;
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374 | RealScalar r0;
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375 | RealScalar lambda, phi, c0, s0;
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376 | MatrixXr l, f;
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377 | // We use the other algorithm which is more efficient for small
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378 | // matrices.
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379 | if (n < algoswap){
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380 | JacobiSVD<MatrixXr> b(m_computed.block(firstCol, firstCol, n + 1, n),
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381 | ComputeFullU | (ComputeFullV * compV)) ;
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382 | if (compU) m_naiveU.block(firstCol, firstCol, n + 1, n + 1).real() << b.matrixU();
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383 | else
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384 | {
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385 | m_naiveU.row(0).segment(firstCol, n + 1).real() << b.matrixU().row(0);
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386 | m_naiveU.row(1).segment(firstCol, n + 1).real() << b.matrixU().row(n);
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387 | }
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388 | if (compV) m_naiveV.block(firstRowW, firstColW, n, n).real() << b.matrixV();
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389 | m_computed.block(firstCol + shift, firstCol + shift, n + 1, n).setZero();
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390 | for (int i=0; i<n; i++)
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391 | {
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392 | m_computed(firstCol + shift + i, firstCol + shift +i) = b.singularValues().coeffRef(i);
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393 | }
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394 | return;
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395 | }
|
---|
396 | // We use the divide and conquer algorithm
|
---|
397 | alphaK = m_computed(firstCol + k, firstCol + k);
|
---|
398 | betaK = m_computed(firstCol + k + 1, firstCol + k);
|
---|
399 | // The divide must be done in that order in order to have good results. Divide change the data inside the submatrices
|
---|
400 | // and the divide of the right submatrice reads one column of the left submatrice. That's why we need to treat the
|
---|
401 | // right submatrix before the left one.
|
---|
402 | divide(k + 1 + firstCol, lastCol, k + 1 + firstRowW, k + 1 + firstColW, shift);
|
---|
403 | divide(firstCol, k - 1 + firstCol, firstRowW, firstColW + 1, shift + 1);
|
---|
404 | if (compU)
|
---|
405 | {
|
---|
406 | lambda = m_naiveU(firstCol + k, firstCol + k);
|
---|
407 | phi = m_naiveU(firstCol + k + 1, lastCol + 1);
|
---|
408 | }
|
---|
409 | else
|
---|
410 | {
|
---|
411 | lambda = m_naiveU(1, firstCol + k);
|
---|
412 | phi = m_naiveU(0, lastCol + 1);
|
---|
413 | }
|
---|
414 | r0 = sqrt((abs(alphaK * lambda) * abs(alphaK * lambda))
|
---|
415 | + abs(betaK * phi) * abs(betaK * phi));
|
---|
416 | if (compU)
|
---|
417 | {
|
---|
418 | l = m_naiveU.row(firstCol + k).segment(firstCol, k);
|
---|
419 | f = m_naiveU.row(firstCol + k + 1).segment(firstCol + k + 1, n - k - 1);
|
---|
420 | }
|
---|
421 | else
|
---|
422 | {
|
---|
423 | l = m_naiveU.row(1).segment(firstCol, k);
|
---|
424 | f = m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1);
|
---|
425 | }
|
---|
426 | if (compV) m_naiveV(firstRowW+k, firstColW) = 1;
|
---|
427 | if (r0 == 0)
|
---|
428 | {
|
---|
429 | c0 = 1;
|
---|
430 | s0 = 0;
|
---|
431 | }
|
---|
432 | else
|
---|
433 | {
|
---|
434 | c0 = alphaK * lambda / r0;
|
---|
435 | s0 = betaK * phi / r0;
|
---|
436 | }
|
---|
437 | if (compU)
|
---|
438 | {
|
---|
439 | MatrixXr q1 (m_naiveU.col(firstCol + k).segment(firstCol, k + 1));
|
---|
440 | // we shiftW Q1 to the right
|
---|
441 | for (Index i = firstCol + k - 1; i >= firstCol; i--)
|
---|
442 | {
|
---|
443 | m_naiveU.col(i + 1).segment(firstCol, k + 1) << m_naiveU.col(i).segment(firstCol, k + 1);
|
---|
444 | }
|
---|
445 | // we shift q1 at the left with a factor c0
|
---|
446 | m_naiveU.col(firstCol).segment( firstCol, k + 1) << (q1 * c0);
|
---|
447 | // last column = q1 * - s0
|
---|
448 | m_naiveU.col(lastCol + 1).segment(firstCol, k + 1) << (q1 * ( - s0));
|
---|
449 | // first column = q2 * s0
|
---|
450 | m_naiveU.col(firstCol).segment(firstCol + k + 1, n - k) <<
|
---|
451 | m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) *s0;
|
---|
452 | // q2 *= c0
|
---|
453 | m_naiveU.col(lastCol + 1).segment(firstCol + k + 1, n - k) *= c0;
|
---|
454 | }
|
---|
455 | else
|
---|
456 | {
|
---|
457 | RealScalar q1 = (m_naiveU(0, firstCol + k));
|
---|
458 | // we shift Q1 to the right
|
---|
459 | for (Index i = firstCol + k - 1; i >= firstCol; i--)
|
---|
460 | {
|
---|
461 | m_naiveU(0, i + 1) = m_naiveU(0, i);
|
---|
462 | }
|
---|
463 | // we shift q1 at the left with a factor c0
|
---|
464 | m_naiveU(0, firstCol) = (q1 * c0);
|
---|
465 | // last column = q1 * - s0
|
---|
466 | m_naiveU(0, lastCol + 1) = (q1 * ( - s0));
|
---|
467 | // first column = q2 * s0
|
---|
468 | m_naiveU(1, firstCol) = m_naiveU(1, lastCol + 1) *s0;
|
---|
469 | // q2 *= c0
|
---|
470 | m_naiveU(1, lastCol + 1) *= c0;
|
---|
471 | m_naiveU.row(1).segment(firstCol + 1, k).setZero();
|
---|
472 | m_naiveU.row(0).segment(firstCol + k + 1, n - k - 1).setZero();
|
---|
473 | }
|
---|
474 | m_computed(firstCol + shift, firstCol + shift) = r0;
|
---|
475 | m_computed.col(firstCol + shift).segment(firstCol + shift + 1, k) << alphaK * l.transpose().real();
|
---|
476 | m_computed.col(firstCol + shift).segment(firstCol + shift + k + 1, n - k - 1) << betaK * f.transpose().real();
|
---|
477 |
|
---|
478 |
|
---|
479 | // the line below do the deflation of the matrix for the third part of the algorithm
|
---|
480 | // Here the deflation is commented because the third part of the algorithm is not implemented
|
---|
481 | // the third part of the algorithm is a fast SVD on the matrix m_computed which works thanks to the deflation
|
---|
482 |
|
---|
483 | deflation(firstCol, lastCol, k, firstRowW, firstColW, shift);
|
---|
484 |
|
---|
485 | // Third part of the algorithm, since the real third part of the algorithm is not implemeted we use a JacobiSVD
|
---|
486 | JacobiSVD<MatrixXr> res= JacobiSVD<MatrixXr>(m_computed.block(firstCol + shift, firstCol +shift, n + 1, n),
|
---|
487 | ComputeFullU | (ComputeFullV * compV)) ;
|
---|
488 | if (compU) m_naiveU.block(firstCol, firstCol, n + 1, n + 1) *= res.matrixU();
|
---|
489 | else m_naiveU.block(0, firstCol, 2, n + 1) *= res.matrixU();
|
---|
490 |
|
---|
491 | if (compV) m_naiveV.block(firstRowW, firstColW, n, n) *= res.matrixV();
|
---|
492 | m_computed.block(firstCol + shift, firstCol + shift, n, n) << MatrixXr::Zero(n, n);
|
---|
493 | for (int i=0; i<n; i++)
|
---|
494 | m_computed(firstCol + shift + i, firstCol + shift +i) = res.singularValues().coeffRef(i);
|
---|
495 | // end of the third part
|
---|
496 |
|
---|
497 |
|
---|
498 | }// end divide
|
---|
499 |
|
---|
500 |
|
---|
501 | // page 12_13
|
---|
502 | // i >= 1, di almost null and zi non null.
|
---|
503 | // We use a rotation to zero out zi applied to the left of M
|
---|
504 | template <typename MatrixType>
|
---|
505 | void BDCSVD<MatrixType>::deflation43(Index firstCol, Index shift, Index i, Index size){
|
---|
506 | using std::abs;
|
---|
507 | using std::sqrt;
|
---|
508 | using std::pow;
|
---|
509 | RealScalar c = m_computed(firstCol + shift, firstCol + shift);
|
---|
510 | RealScalar s = m_computed(i, firstCol + shift);
|
---|
511 | RealScalar r = sqrt(pow(abs(c), 2) + pow(abs(s), 2));
|
---|
512 | if (r == 0){
|
---|
513 | m_computed(i, i)=0;
|
---|
514 | return;
|
---|
515 | }
|
---|
516 | c/=r;
|
---|
517 | s/=r;
|
---|
518 | m_computed(firstCol + shift, firstCol + shift) = r;
|
---|
519 | m_computed(i, firstCol + shift) = 0;
|
---|
520 | m_computed(i, i) = 0;
|
---|
521 | if (compU){
|
---|
522 | m_naiveU.col(firstCol).segment(firstCol,size) =
|
---|
523 | c * m_naiveU.col(firstCol).segment(firstCol, size) -
|
---|
524 | s * m_naiveU.col(i).segment(firstCol, size) ;
|
---|
525 |
|
---|
526 | m_naiveU.col(i).segment(firstCol, size) =
|
---|
527 | (c + s*s/c) * m_naiveU.col(i).segment(firstCol, size) +
|
---|
528 | (s/c) * m_naiveU.col(firstCol).segment(firstCol,size);
|
---|
529 | }
|
---|
530 | }// end deflation 43
|
---|
531 |
|
---|
532 |
|
---|
533 | // page 13
|
---|
534 | // i,j >= 1, i != j and |di - dj| < epsilon * norm2(M)
|
---|
535 | // We apply two rotations to have zj = 0;
|
---|
536 | template <typename MatrixType>
|
---|
537 | void BDCSVD<MatrixType>::deflation44(Index firstColu , Index firstColm, Index firstRowW, Index firstColW, Index i, Index j, Index size){
|
---|
538 | using std::abs;
|
---|
539 | using std::sqrt;
|
---|
540 | using std::conj;
|
---|
541 | using std::pow;
|
---|
542 | RealScalar c = m_computed(firstColm, firstColm + j - 1);
|
---|
543 | RealScalar s = m_computed(firstColm, firstColm + i - 1);
|
---|
544 | RealScalar r = sqrt(pow(abs(c), 2) + pow(abs(s), 2));
|
---|
545 | if (r==0){
|
---|
546 | m_computed(firstColm + i, firstColm + i) = m_computed(firstColm + j, firstColm + j);
|
---|
547 | return;
|
---|
548 | }
|
---|
549 | c/=r;
|
---|
550 | s/=r;
|
---|
551 | m_computed(firstColm + i, firstColm) = r;
|
---|
552 | m_computed(firstColm + i, firstColm + i) = m_computed(firstColm + j, firstColm + j);
|
---|
553 | m_computed(firstColm + j, firstColm) = 0;
|
---|
554 | if (compU){
|
---|
555 | m_naiveU.col(firstColu + i).segment(firstColu, size) =
|
---|
556 | c * m_naiveU.col(firstColu + i).segment(firstColu, size) -
|
---|
557 | s * m_naiveU.col(firstColu + j).segment(firstColu, size) ;
|
---|
558 |
|
---|
559 | m_naiveU.col(firstColu + j).segment(firstColu, size) =
|
---|
560 | (c + s*s/c) * m_naiveU.col(firstColu + j).segment(firstColu, size) +
|
---|
561 | (s/c) * m_naiveU.col(firstColu + i).segment(firstColu, size);
|
---|
562 | }
|
---|
563 | if (compV){
|
---|
564 | m_naiveV.col(firstColW + i).segment(firstRowW, size - 1) =
|
---|
565 | c * m_naiveV.col(firstColW + i).segment(firstRowW, size - 1) +
|
---|
566 | s * m_naiveV.col(firstColW + j).segment(firstRowW, size - 1) ;
|
---|
567 |
|
---|
568 | m_naiveV.col(firstColW + j).segment(firstRowW, size - 1) =
|
---|
569 | (c + s*s/c) * m_naiveV.col(firstColW + j).segment(firstRowW, size - 1) -
|
---|
570 | (s/c) * m_naiveV.col(firstColW + i).segment(firstRowW, size - 1);
|
---|
571 | }
|
---|
572 | }// end deflation 44
|
---|
573 |
|
---|
574 |
|
---|
575 |
|
---|
576 | template <typename MatrixType>
|
---|
577 | void BDCSVD<MatrixType>::deflation(Index firstCol, Index lastCol, Index k, Index firstRowW, Index firstColW, Index shift){
|
---|
578 | //condition 4.1
|
---|
579 | RealScalar EPS = EPSILON * (std::max<RealScalar>(m_computed(firstCol + shift + 1, firstCol + shift + 1), m_computed(firstCol + k, firstCol + k)));
|
---|
580 | const Index length = lastCol + 1 - firstCol;
|
---|
581 | if (m_computed(firstCol + shift, firstCol + shift) < EPS){
|
---|
582 | m_computed(firstCol + shift, firstCol + shift) = EPS;
|
---|
583 | }
|
---|
584 | //condition 4.2
|
---|
585 | for (Index i=firstCol + shift + 1;i<=lastCol + shift;i++){
|
---|
586 | if (std::abs(m_computed(i, firstCol + shift)) < EPS){
|
---|
587 | m_computed(i, firstCol + shift) = 0;
|
---|
588 | }
|
---|
589 | }
|
---|
590 |
|
---|
591 | //condition 4.3
|
---|
592 | for (Index i=firstCol + shift + 1;i<=lastCol + shift; i++){
|
---|
593 | if (m_computed(i, i) < EPS){
|
---|
594 | deflation43(firstCol, shift, i, length);
|
---|
595 | }
|
---|
596 | }
|
---|
597 |
|
---|
598 | //condition 4.4
|
---|
599 |
|
---|
600 | Index i=firstCol + shift + 1, j=firstCol + shift + k + 1;
|
---|
601 | //we stock the final place of each line
|
---|
602 | Index *permutation = new Index[length];
|
---|
603 |
|
---|
604 | for (Index p =1; p < length; p++) {
|
---|
605 | if (i> firstCol + shift + k){
|
---|
606 | permutation[p] = j;
|
---|
607 | j++;
|
---|
608 | } else if (j> lastCol + shift)
|
---|
609 | {
|
---|
610 | permutation[p] = i;
|
---|
611 | i++;
|
---|
612 | }
|
---|
613 | else
|
---|
614 | {
|
---|
615 | if (m_computed(i, i) < m_computed(j, j)){
|
---|
616 | permutation[p] = j;
|
---|
617 | j++;
|
---|
618 | }
|
---|
619 | else
|
---|
620 | {
|
---|
621 | permutation[p] = i;
|
---|
622 | i++;
|
---|
623 | }
|
---|
624 | }
|
---|
625 | }
|
---|
626 | //we do the permutation
|
---|
627 | RealScalar aux;
|
---|
628 | //we stock the current index of each col
|
---|
629 | //and the column of each index
|
---|
630 | Index *realInd = new Index[length];
|
---|
631 | Index *realCol = new Index[length];
|
---|
632 | for (int pos = 0; pos< length; pos++){
|
---|
633 | realCol[pos] = pos + firstCol + shift;
|
---|
634 | realInd[pos] = pos;
|
---|
635 | }
|
---|
636 | const Index Zero = firstCol + shift;
|
---|
637 | VectorType temp;
|
---|
638 | for (int i = 1; i < length - 1; i++){
|
---|
639 | const Index I = i + Zero;
|
---|
640 | const Index realI = realInd[i];
|
---|
641 | const Index j = permutation[length - i] - Zero;
|
---|
642 | const Index J = realCol[j];
|
---|
643 |
|
---|
644 | //diag displace
|
---|
645 | aux = m_computed(I, I);
|
---|
646 | m_computed(I, I) = m_computed(J, J);
|
---|
647 | m_computed(J, J) = aux;
|
---|
648 |
|
---|
649 | //firstrow displace
|
---|
650 | aux = m_computed(I, Zero);
|
---|
651 | m_computed(I, Zero) = m_computed(J, Zero);
|
---|
652 | m_computed(J, Zero) = aux;
|
---|
653 |
|
---|
654 | // change columns
|
---|
655 | if (compU) {
|
---|
656 | temp = m_naiveU.col(I - shift).segment(firstCol, length + 1);
|
---|
657 | m_naiveU.col(I - shift).segment(firstCol, length + 1) <<
|
---|
658 | m_naiveU.col(J - shift).segment(firstCol, length + 1);
|
---|
659 | m_naiveU.col(J - shift).segment(firstCol, length + 1) << temp;
|
---|
660 | }
|
---|
661 | else
|
---|
662 | {
|
---|
663 | temp = m_naiveU.col(I - shift).segment(0, 2);
|
---|
664 | m_naiveU.col(I - shift).segment(0, 2) <<
|
---|
665 | m_naiveU.col(J - shift).segment(0, 2);
|
---|
666 | m_naiveU.col(J - shift).segment(0, 2) << temp;
|
---|
667 | }
|
---|
668 | if (compV) {
|
---|
669 | const Index CWI = I + firstColW - Zero;
|
---|
670 | const Index CWJ = J + firstColW - Zero;
|
---|
671 | temp = m_naiveV.col(CWI).segment(firstRowW, length);
|
---|
672 | m_naiveV.col(CWI).segment(firstRowW, length) << m_naiveV.col(CWJ).segment(firstRowW, length);
|
---|
673 | m_naiveV.col(CWJ).segment(firstRowW, length) << temp;
|
---|
674 | }
|
---|
675 |
|
---|
676 | //update real pos
|
---|
677 | realCol[realI] = J;
|
---|
678 | realCol[j] = I;
|
---|
679 | realInd[J - Zero] = realI;
|
---|
680 | realInd[I - Zero] = j;
|
---|
681 | }
|
---|
682 | for (Index i = firstCol + shift + 1; i<lastCol + shift;i++){
|
---|
683 | if ((m_computed(i + 1, i + 1) - m_computed(i, i)) < EPS){
|
---|
684 | deflation44(firstCol ,
|
---|
685 | firstCol + shift,
|
---|
686 | firstRowW,
|
---|
687 | firstColW,
|
---|
688 | i - Zero,
|
---|
689 | i + 1 - Zero,
|
---|
690 | length);
|
---|
691 | }
|
---|
692 | }
|
---|
693 | delete [] permutation;
|
---|
694 | delete [] realInd;
|
---|
695 | delete [] realCol;
|
---|
696 |
|
---|
697 | }//end deflation
|
---|
698 |
|
---|
699 |
|
---|
700 | namespace internal{
|
---|
701 |
|
---|
702 | template<typename _MatrixType, typename Rhs>
|
---|
703 | struct solve_retval<BDCSVD<_MatrixType>, Rhs>
|
---|
704 | : solve_retval_base<BDCSVD<_MatrixType>, Rhs>
|
---|
705 | {
|
---|
706 | typedef BDCSVD<_MatrixType> BDCSVDType;
|
---|
707 | EIGEN_MAKE_SOLVE_HELPERS(BDCSVDType, Rhs)
|
---|
708 |
|
---|
709 | template<typename Dest> void evalTo(Dest& dst) const
|
---|
710 | {
|
---|
711 | eigen_assert(rhs().rows() == dec().rows());
|
---|
712 | // A = U S V^*
|
---|
713 | // So A^{ - 1} = V S^{ - 1} U^*
|
---|
714 | Index diagSize = (std::min)(dec().rows(), dec().cols());
|
---|
715 | typename BDCSVDType::SingularValuesType invertedSingVals(diagSize);
|
---|
716 | Index nonzeroSingVals = dec().nonzeroSingularValues();
|
---|
717 | invertedSingVals.head(nonzeroSingVals) = dec().singularValues().head(nonzeroSingVals).array().inverse();
|
---|
718 | invertedSingVals.tail(diagSize - nonzeroSingVals).setZero();
|
---|
719 |
|
---|
720 | dst = dec().matrixV().leftCols(diagSize)
|
---|
721 | * invertedSingVals.asDiagonal()
|
---|
722 | * dec().matrixU().leftCols(diagSize).adjoint()
|
---|
723 | * rhs();
|
---|
724 | return;
|
---|
725 | }
|
---|
726 | };
|
---|
727 |
|
---|
728 | } //end namespace internal
|
---|
729 |
|
---|
730 | /** \svd_module
|
---|
731 | *
|
---|
732 | * \return the singular value decomposition of \c *this computed by
|
---|
733 | * BDC Algorithm
|
---|
734 | *
|
---|
735 | * \sa class BDCSVD
|
---|
736 | */
|
---|
737 | /*
|
---|
738 | template<typename Derived>
|
---|
739 | BDCSVD<typename MatrixBase<Derived>::PlainObject>
|
---|
740 | MatrixBase<Derived>::bdcSvd(unsigned int computationOptions) const
|
---|
741 | {
|
---|
742 | return BDCSVD<PlainObject>(*this, computationOptions);
|
---|
743 | }
|
---|
744 | */
|
---|
745 |
|
---|
746 | } // end namespace Eigen
|
---|
747 |
|
---|
748 | #endif
|
---|