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) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
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5 | // Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
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6 | //
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7 | // This Source Code Form is subject to the terms of the Mozilla
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8 | // Public License v. 2.0. If a copy of the MPL was not distributed
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9 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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10 |
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11 | #ifndef EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
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12 | #define EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
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13 |
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14 | namespace Eigen {
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15 |
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16 | /** \ingroup QR_Module
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17 | *
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18 | * \class ColPivHouseholderQR
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19 | *
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20 | * \brief Householder rank-revealing QR decomposition of a matrix with column-pivoting
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21 | *
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22 | * \param MatrixType the type of the matrix of which we are computing the QR decomposition
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23 | *
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24 | * This class performs a rank-revealing QR decomposition of a matrix \b A into matrices \b P, \b Q and \b R
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25 | * such that
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26 | * \f[
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27 | * \mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \mathbf{R}
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28 | * \f]
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29 | * by using Householder transformations. Here, \b P is a permutation matrix, \b Q a unitary matrix and \b R an
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30 | * upper triangular matrix.
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31 | *
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32 | * This decomposition performs column pivoting in order to be rank-revealing and improve
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33 | * numerical stability. It is slower than HouseholderQR, and faster than FullPivHouseholderQR.
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34 | *
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35 | * \sa MatrixBase::colPivHouseholderQr()
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36 | */
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37 | template<typename _MatrixType> class ColPivHouseholderQR
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38 | {
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39 | public:
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40 |
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41 | typedef _MatrixType MatrixType;
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42 | enum {
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43 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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44 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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45 | Options = MatrixType::Options,
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46 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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47 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
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48 | };
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49 | typedef typename MatrixType::Scalar Scalar;
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50 | typedef typename MatrixType::RealScalar RealScalar;
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51 | typedef typename MatrixType::Index Index;
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52 | typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, Options, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixQType;
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53 | typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
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54 | typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType;
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55 | typedef typename internal::plain_row_type<MatrixType, Index>::type IntRowVectorType;
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56 | typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
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57 | typedef typename internal::plain_row_type<MatrixType, RealScalar>::type RealRowVectorType;
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58 | typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type> HouseholderSequenceType;
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59 |
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60 | private:
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61 |
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62 | typedef typename PermutationType::Index PermIndexType;
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63 |
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64 | public:
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65 |
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66 | /**
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67 | * \brief Default Constructor.
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68 | *
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69 | * The default constructor is useful in cases in which the user intends to
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70 | * perform decompositions via ColPivHouseholderQR::compute(const MatrixType&).
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71 | */
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72 | ColPivHouseholderQR()
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73 | : m_qr(),
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74 | m_hCoeffs(),
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75 | m_colsPermutation(),
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76 | m_colsTranspositions(),
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77 | m_temp(),
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78 | m_colSqNorms(),
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79 | m_isInitialized(false),
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80 | m_usePrescribedThreshold(false) {}
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81 |
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82 | /** \brief Default Constructor with memory preallocation
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83 | *
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84 | * Like the default constructor but with preallocation of the internal data
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85 | * according to the specified problem \a size.
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86 | * \sa ColPivHouseholderQR()
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87 | */
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88 | ColPivHouseholderQR(Index rows, Index cols)
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89 | : m_qr(rows, cols),
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90 | m_hCoeffs((std::min)(rows,cols)),
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91 | m_colsPermutation(PermIndexType(cols)),
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92 | m_colsTranspositions(cols),
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93 | m_temp(cols),
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94 | m_colSqNorms(cols),
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95 | m_isInitialized(false),
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96 | m_usePrescribedThreshold(false) {}
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97 |
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98 | /** \brief Constructs a QR factorization from a given matrix
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99 | *
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100 | * This constructor computes the QR factorization of the matrix \a matrix by calling
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101 | * the method compute(). It is a short cut for:
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102 | *
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103 | * \code
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104 | * ColPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
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105 | * qr.compute(matrix);
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106 | * \endcode
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107 | *
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108 | * \sa compute()
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109 | */
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110 | ColPivHouseholderQR(const MatrixType& matrix)
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111 | : m_qr(matrix.rows(), matrix.cols()),
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112 | m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
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113 | m_colsPermutation(PermIndexType(matrix.cols())),
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114 | m_colsTranspositions(matrix.cols()),
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115 | m_temp(matrix.cols()),
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116 | m_colSqNorms(matrix.cols()),
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117 | m_isInitialized(false),
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118 | m_usePrescribedThreshold(false)
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119 | {
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120 | compute(matrix);
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121 | }
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122 |
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123 | /** This method finds a solution x to the equation Ax=b, where A is the matrix of which
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124 | * *this is the QR decomposition, if any exists.
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125 | *
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126 | * \param b the right-hand-side of the equation to solve.
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127 | *
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128 | * \returns a solution.
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129 | *
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130 | * \note The case where b is a matrix is not yet implemented. Also, this
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131 | * code is space inefficient.
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132 | *
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133 | * \note_about_checking_solutions
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134 | *
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135 | * \note_about_arbitrary_choice_of_solution
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136 | *
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137 | * Example: \include ColPivHouseholderQR_solve.cpp
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138 | * Output: \verbinclude ColPivHouseholderQR_solve.out
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139 | */
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140 | template<typename Rhs>
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141 | inline const internal::solve_retval<ColPivHouseholderQR, Rhs>
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142 | solve(const MatrixBase<Rhs>& b) const
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143 | {
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144 | eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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145 | return internal::solve_retval<ColPivHouseholderQR, Rhs>(*this, b.derived());
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146 | }
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147 |
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148 | HouseholderSequenceType householderQ(void) const;
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149 | HouseholderSequenceType matrixQ(void) const
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150 | {
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151 | return householderQ();
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152 | }
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153 |
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154 | /** \returns a reference to the matrix where the Householder QR decomposition is stored
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155 | */
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156 | const MatrixType& matrixQR() const
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157 | {
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158 | eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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159 | return m_qr;
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160 | }
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161 |
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162 | /** \returns a reference to the matrix where the result Householder QR is stored
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163 | * \warning The strict lower part of this matrix contains internal values.
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164 | * Only the upper triangular part should be referenced. To get it, use
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165 | * \code matrixR().template triangularView<Upper>() \endcode
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166 | * For rank-deficient matrices, use
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167 | * \code
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168 | * matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()
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169 | * \endcode
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170 | */
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171 | const MatrixType& matrixR() const
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172 | {
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173 | eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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174 | return m_qr;
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175 | }
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176 |
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177 | ColPivHouseholderQR& compute(const MatrixType& matrix);
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178 |
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179 | /** \returns a const reference to the column permutation matrix */
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180 | const PermutationType& colsPermutation() const
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181 | {
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182 | eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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183 | return m_colsPermutation;
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184 | }
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185 |
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186 | /** \returns the absolute value of the determinant of the matrix of which
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187 | * *this is the QR decomposition. It has only linear complexity
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188 | * (that is, O(n) where n is the dimension of the square matrix)
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189 | * as the QR decomposition has already been computed.
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190 | *
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191 | * \note This is only for square matrices.
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192 | *
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193 | * \warning a determinant can be very big or small, so for matrices
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194 | * of large enough dimension, there is a risk of overflow/underflow.
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195 | * One way to work around that is to use logAbsDeterminant() instead.
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196 | *
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197 | * \sa logAbsDeterminant(), MatrixBase::determinant()
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198 | */
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199 | typename MatrixType::RealScalar absDeterminant() const;
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200 |
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201 | /** \returns the natural log of the absolute value of the determinant of the matrix of which
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202 | * *this is the QR decomposition. It has only linear complexity
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203 | * (that is, O(n) where n is the dimension of the square matrix)
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204 | * as the QR decomposition has already been computed.
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205 | *
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206 | * \note This is only for square matrices.
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207 | *
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208 | * \note This method is useful to work around the risk of overflow/underflow that's inherent
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209 | * to determinant computation.
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210 | *
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211 | * \sa absDeterminant(), MatrixBase::determinant()
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212 | */
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213 | typename MatrixType::RealScalar logAbsDeterminant() const;
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214 |
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215 | /** \returns the rank of the matrix of which *this is the QR decomposition.
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216 | *
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217 | * \note This method has to determine which pivots should be considered nonzero.
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218 | * For that, it uses the threshold value that you can control by calling
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219 | * setThreshold(const RealScalar&).
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220 | */
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221 | inline Index rank() const
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222 | {
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223 | using std::abs;
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224 | eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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225 | RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
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226 | Index result = 0;
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227 | for(Index i = 0; i < m_nonzero_pivots; ++i)
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228 | result += (abs(m_qr.coeff(i,i)) > premultiplied_threshold);
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229 | return result;
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230 | }
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231 |
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232 | /** \returns the dimension of the kernel of the matrix of which *this is the QR decomposition.
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233 | *
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234 | * \note This method has to determine which pivots should be considered nonzero.
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235 | * For that, it uses the threshold value that you can control by calling
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236 | * setThreshold(const RealScalar&).
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237 | */
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238 | inline Index dimensionOfKernel() const
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239 | {
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240 | eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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241 | return cols() - rank();
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242 | }
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243 |
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244 | /** \returns true if the matrix of which *this is the QR decomposition represents an injective
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245 | * linear map, i.e. has trivial kernel; false otherwise.
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246 | *
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247 | * \note This method has to determine which pivots should be considered nonzero.
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248 | * For that, it uses the threshold value that you can control by calling
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249 | * setThreshold(const RealScalar&).
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250 | */
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251 | inline bool isInjective() const
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252 | {
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253 | eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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254 | return rank() == cols();
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255 | }
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256 |
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257 | /** \returns true if the matrix of which *this is the QR decomposition represents a surjective
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258 | * linear map; false otherwise.
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259 | *
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260 | * \note This method has to determine which pivots should be considered nonzero.
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261 | * For that, it uses the threshold value that you can control by calling
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262 | * setThreshold(const RealScalar&).
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263 | */
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264 | inline bool isSurjective() const
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265 | {
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266 | eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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267 | return rank() == rows();
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268 | }
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269 |
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270 | /** \returns true if the matrix of which *this is the QR decomposition is invertible.
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271 | *
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272 | * \note This method has to determine which pivots should be considered nonzero.
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273 | * For that, it uses the threshold value that you can control by calling
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274 | * setThreshold(const RealScalar&).
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275 | */
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276 | inline bool isInvertible() const
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277 | {
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278 | eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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279 | return isInjective() && isSurjective();
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280 | }
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281 |
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282 | /** \returns the inverse of the matrix of which *this is the QR decomposition.
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283 | *
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284 | * \note If this matrix is not invertible, the returned matrix has undefined coefficients.
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285 | * Use isInvertible() to first determine whether this matrix is invertible.
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286 | */
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287 | inline const
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288 | internal::solve_retval<ColPivHouseholderQR, typename MatrixType::IdentityReturnType>
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289 | inverse() const
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290 | {
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291 | eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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292 | return internal::solve_retval<ColPivHouseholderQR,typename MatrixType::IdentityReturnType>
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293 | (*this, MatrixType::Identity(m_qr.rows(), m_qr.cols()));
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294 | }
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295 |
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296 | inline Index rows() const { return m_qr.rows(); }
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297 | inline Index cols() const { return m_qr.cols(); }
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298 |
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299 | /** \returns a const reference to the vector of Householder coefficients used to represent the factor \c Q.
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300 | *
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301 | * For advanced uses only.
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302 | */
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303 | const HCoeffsType& hCoeffs() const { return m_hCoeffs; }
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304 |
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305 | /** Allows to prescribe a threshold to be used by certain methods, such as rank(),
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306 | * who need to determine when pivots are to be considered nonzero. This is not used for the
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307 | * QR decomposition itself.
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308 | *
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309 | * When it needs to get the threshold value, Eigen calls threshold(). By default, this
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310 | * uses a formula to automatically determine a reasonable threshold.
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311 | * Once you have called the present method setThreshold(const RealScalar&),
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312 | * your value is used instead.
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313 | *
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314 | * \param threshold The new value to use as the threshold.
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315 | *
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316 | * A pivot will be considered nonzero if its absolute value is strictly greater than
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317 | * \f$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \f$
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318 | * where maxpivot is the biggest pivot.
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319 | *
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320 | * If you want to come back to the default behavior, call setThreshold(Default_t)
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321 | */
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322 | ColPivHouseholderQR& setThreshold(const RealScalar& threshold)
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323 | {
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324 | m_usePrescribedThreshold = true;
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325 | m_prescribedThreshold = threshold;
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326 | return *this;
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327 | }
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328 |
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329 | /** Allows to come back to the default behavior, letting Eigen use its default formula for
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330 | * determining the threshold.
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331 | *
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332 | * You should pass the special object Eigen::Default as parameter here.
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333 | * \code qr.setThreshold(Eigen::Default); \endcode
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334 | *
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335 | * See the documentation of setThreshold(const RealScalar&).
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336 | */
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337 | ColPivHouseholderQR& setThreshold(Default_t)
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338 | {
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339 | m_usePrescribedThreshold = false;
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340 | return *this;
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341 | }
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342 |
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343 | /** Returns the threshold that will be used by certain methods such as rank().
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344 | *
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345 | * See the documentation of setThreshold(const RealScalar&).
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346 | */
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347 | RealScalar threshold() const
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348 | {
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349 | eigen_assert(m_isInitialized || m_usePrescribedThreshold);
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350 | return m_usePrescribedThreshold ? m_prescribedThreshold
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351 | // this formula comes from experimenting (see "LU precision tuning" thread on the list)
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352 | // and turns out to be identical to Higham's formula used already in LDLt.
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353 | : NumTraits<Scalar>::epsilon() * RealScalar(m_qr.diagonalSize());
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354 | }
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355 |
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356 | /** \returns the number of nonzero pivots in the QR decomposition.
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357 | * Here nonzero is meant in the exact sense, not in a fuzzy sense.
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358 | * So that notion isn't really intrinsically interesting, but it is
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359 | * still useful when implementing algorithms.
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360 | *
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361 | * \sa rank()
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362 | */
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363 | inline Index nonzeroPivots() const
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364 | {
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365 | eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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366 | return m_nonzero_pivots;
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367 | }
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368 |
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369 | /** \returns the absolute value of the biggest pivot, i.e. the biggest
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370 | * diagonal coefficient of R.
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371 | */
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372 | RealScalar maxPivot() const { return m_maxpivot; }
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373 |
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374 | /** \brief Reports whether the QR factorization was succesful.
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375 | *
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376 | * \note This function always returns \c Success. It is provided for compatibility
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377 | * with other factorization routines.
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378 | * \returns \c Success
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379 | */
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380 | ComputationInfo info() const
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381 | {
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382 | eigen_assert(m_isInitialized && "Decomposition is not initialized.");
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383 | return Success;
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384 | }
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385 |
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386 | protected:
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387 |
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388 | static void check_template_parameters()
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389 | {
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390 | EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
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391 | }
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392 |
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393 | MatrixType m_qr;
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394 | HCoeffsType m_hCoeffs;
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395 | PermutationType m_colsPermutation;
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396 | IntRowVectorType m_colsTranspositions;
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397 | RowVectorType m_temp;
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398 | RealRowVectorType m_colSqNorms;
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399 | bool m_isInitialized, m_usePrescribedThreshold;
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400 | RealScalar m_prescribedThreshold, m_maxpivot;
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401 | Index m_nonzero_pivots;
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402 | Index m_det_pq;
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403 | };
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404 |
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405 | template<typename MatrixType>
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406 | typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::absDeterminant() const
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407 | {
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408 | using std::abs;
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409 | eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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410 | eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
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411 | return abs(m_qr.diagonal().prod());
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412 | }
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413 |
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414 | template<typename MatrixType>
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415 | typename MatrixType::RealScalar ColPivHouseholderQR<MatrixType>::logAbsDeterminant() const
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416 | {
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417 | eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
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418 | eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
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419 | return m_qr.diagonal().cwiseAbs().array().log().sum();
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420 | }
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421 |
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422 | /** Performs the QR factorization of the given matrix \a matrix. The result of
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423 | * the factorization is stored into \c *this, and a reference to \c *this
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424 | * is returned.
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425 | *
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426 | * \sa class ColPivHouseholderQR, ColPivHouseholderQR(const MatrixType&)
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427 | */
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428 | template<typename MatrixType>
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429 | ColPivHouseholderQR<MatrixType>& ColPivHouseholderQR<MatrixType>::compute(const MatrixType& matrix)
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430 | {
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431 | check_template_parameters();
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432 |
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433 | using std::abs;
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434 | Index rows = matrix.rows();
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435 | Index cols = matrix.cols();
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436 | Index size = matrix.diagonalSize();
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437 |
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438 | // the column permutation is stored as int indices, so just to be sure:
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439 | eigen_assert(cols<=NumTraits<int>::highest());
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440 |
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441 | m_qr = matrix;
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442 | m_hCoeffs.resize(size);
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443 |
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444 | m_temp.resize(cols);
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445 |
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446 | m_colsTranspositions.resize(matrix.cols());
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447 | Index number_of_transpositions = 0;
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448 |
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449 | m_colSqNorms.resize(cols);
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450 | for(Index k = 0; k < cols; ++k)
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451 | m_colSqNorms.coeffRef(k) = m_qr.col(k).squaredNorm();
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452 |
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453 | RealScalar threshold_helper = m_colSqNorms.maxCoeff() * numext::abs2(NumTraits<Scalar>::epsilon()) / RealScalar(rows);
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454 |
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455 | m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
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456 | m_maxpivot = RealScalar(0);
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457 |
|
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458 | for(Index k = 0; k < size; ++k)
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459 | {
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460 | // first, we look up in our table m_colSqNorms which column has the biggest squared norm
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461 | Index biggest_col_index;
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462 | RealScalar biggest_col_sq_norm = m_colSqNorms.tail(cols-k).maxCoeff(&biggest_col_index);
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463 | biggest_col_index += k;
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464 |
|
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465 | // since our table m_colSqNorms accumulates imprecision at every step, we must now recompute
|
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466 | // the actual squared norm of the selected column.
|
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467 | // Note that not doing so does result in solve() sometimes returning inf/nan values
|
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468 | // when running the unit test with 1000 repetitions.
|
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469 | biggest_col_sq_norm = m_qr.col(biggest_col_index).tail(rows-k).squaredNorm();
|
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470 |
|
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471 | // we store that back into our table: it can't hurt to correct our table.
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472 | m_colSqNorms.coeffRef(biggest_col_index) = biggest_col_sq_norm;
|
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473 |
|
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474 | // Track the number of meaningful pivots but do not stop the decomposition to make
|
---|
475 | // sure that the initial matrix is properly reproduced. See bug 941.
|
---|
476 | if(m_nonzero_pivots==size && biggest_col_sq_norm < threshold_helper * RealScalar(rows-k))
|
---|
477 | m_nonzero_pivots = k;
|
---|
478 |
|
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479 | // apply the transposition to the columns
|
---|
480 | m_colsTranspositions.coeffRef(k) = biggest_col_index;
|
---|
481 | if(k != biggest_col_index) {
|
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482 | m_qr.col(k).swap(m_qr.col(biggest_col_index));
|
---|
483 | std::swap(m_colSqNorms.coeffRef(k), m_colSqNorms.coeffRef(biggest_col_index));
|
---|
484 | ++number_of_transpositions;
|
---|
485 | }
|
---|
486 |
|
---|
487 | // generate the householder vector, store it below the diagonal
|
---|
488 | RealScalar beta;
|
---|
489 | m_qr.col(k).tail(rows-k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
|
---|
490 |
|
---|
491 | // apply the householder transformation to the diagonal coefficient
|
---|
492 | m_qr.coeffRef(k,k) = beta;
|
---|
493 |
|
---|
494 | // remember the maximum absolute value of diagonal coefficients
|
---|
495 | if(abs(beta) > m_maxpivot) m_maxpivot = abs(beta);
|
---|
496 |
|
---|
497 | // apply the householder transformation
|
---|
498 | m_qr.bottomRightCorner(rows-k, cols-k-1)
|
---|
499 | .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), m_hCoeffs.coeffRef(k), &m_temp.coeffRef(k+1));
|
---|
500 |
|
---|
501 | // update our table of squared norms of the columns
|
---|
502 | m_colSqNorms.tail(cols-k-1) -= m_qr.row(k).tail(cols-k-1).cwiseAbs2();
|
---|
503 | }
|
---|
504 |
|
---|
505 | m_colsPermutation.setIdentity(PermIndexType(cols));
|
---|
506 | for(PermIndexType k = 0; k < size/*m_nonzero_pivots*/; ++k)
|
---|
507 | m_colsPermutation.applyTranspositionOnTheRight(k, PermIndexType(m_colsTranspositions.coeff(k)));
|
---|
508 |
|
---|
509 | m_det_pq = (number_of_transpositions%2) ? -1 : 1;
|
---|
510 | m_isInitialized = true;
|
---|
511 |
|
---|
512 | return *this;
|
---|
513 | }
|
---|
514 |
|
---|
515 | namespace internal {
|
---|
516 |
|
---|
517 | template<typename _MatrixType, typename Rhs>
|
---|
518 | struct solve_retval<ColPivHouseholderQR<_MatrixType>, Rhs>
|
---|
519 | : solve_retval_base<ColPivHouseholderQR<_MatrixType>, Rhs>
|
---|
520 | {
|
---|
521 | EIGEN_MAKE_SOLVE_HELPERS(ColPivHouseholderQR<_MatrixType>,Rhs)
|
---|
522 |
|
---|
523 | template<typename Dest> void evalTo(Dest& dst) const
|
---|
524 | {
|
---|
525 | eigen_assert(rhs().rows() == dec().rows());
|
---|
526 |
|
---|
527 | const Index cols = dec().cols(),
|
---|
528 | nonzero_pivots = dec().nonzeroPivots();
|
---|
529 |
|
---|
530 | if(nonzero_pivots == 0)
|
---|
531 | {
|
---|
532 | dst.setZero();
|
---|
533 | return;
|
---|
534 | }
|
---|
535 |
|
---|
536 | typename Rhs::PlainObject c(rhs());
|
---|
537 |
|
---|
538 | // Note that the matrix Q = H_0^* H_1^*... so its inverse is Q^* = (H_0 H_1 ...)^T
|
---|
539 | c.applyOnTheLeft(householderSequence(dec().matrixQR(), dec().hCoeffs())
|
---|
540 | .setLength(dec().nonzeroPivots())
|
---|
541 | .transpose()
|
---|
542 | );
|
---|
543 |
|
---|
544 | dec().matrixR()
|
---|
545 | .topLeftCorner(nonzero_pivots, nonzero_pivots)
|
---|
546 | .template triangularView<Upper>()
|
---|
547 | .solveInPlace(c.topRows(nonzero_pivots));
|
---|
548 |
|
---|
549 | for(Index i = 0; i < nonzero_pivots; ++i) dst.row(dec().colsPermutation().indices().coeff(i)) = c.row(i);
|
---|
550 | for(Index i = nonzero_pivots; i < cols; ++i) dst.row(dec().colsPermutation().indices().coeff(i)).setZero();
|
---|
551 | }
|
---|
552 | };
|
---|
553 |
|
---|
554 | } // end namespace internal
|
---|
555 |
|
---|
556 | /** \returns the matrix Q as a sequence of householder transformations.
|
---|
557 | * You can extract the meaningful part only by using:
|
---|
558 | * \code qr.householderQ().setLength(qr.nonzeroPivots()) \endcode*/
|
---|
559 | template<typename MatrixType>
|
---|
560 | typename ColPivHouseholderQR<MatrixType>::HouseholderSequenceType ColPivHouseholderQR<MatrixType>
|
---|
561 | ::householderQ() const
|
---|
562 | {
|
---|
563 | eigen_assert(m_isInitialized && "ColPivHouseholderQR is not initialized.");
|
---|
564 | return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
|
---|
565 | }
|
---|
566 |
|
---|
567 | /** \return the column-pivoting Householder QR decomposition of \c *this.
|
---|
568 | *
|
---|
569 | * \sa class ColPivHouseholderQR
|
---|
570 | */
|
---|
571 | template<typename Derived>
|
---|
572 | const ColPivHouseholderQR<typename MatrixBase<Derived>::PlainObject>
|
---|
573 | MatrixBase<Derived>::colPivHouseholderQr() const
|
---|
574 | {
|
---|
575 | return ColPivHouseholderQR<PlainObject>(eval());
|
---|
576 | }
|
---|
577 |
|
---|
578 | } // end namespace Eigen
|
---|
579 |
|
---|
580 | #endif // EIGEN_COLPIVOTINGHOUSEHOLDERQR_H
|
---|