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_FULLPIVOTINGHOUSEHOLDERQR_H
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12 | #define EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
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13 |
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14 | namespace Eigen {
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15 |
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16 | namespace internal {
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17 |
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18 | template<typename MatrixType> struct FullPivHouseholderQRMatrixQReturnType;
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19 |
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20 | template<typename MatrixType>
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21 | struct traits<FullPivHouseholderQRMatrixQReturnType<MatrixType> >
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22 | {
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23 | typedef typename MatrixType::PlainObject ReturnType;
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24 | };
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25 |
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26 | }
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27 |
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28 | /** \ingroup QR_Module
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29 | *
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30 | * \class FullPivHouseholderQR
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31 | *
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32 | * \brief Householder rank-revealing QR decomposition of a matrix with full pivoting
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33 | *
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34 | * \param MatrixType the type of the matrix of which we are computing the QR decomposition
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35 | *
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36 | * 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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37 | * such that
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38 | * \f[
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39 | * \mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \mathbf{R}
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40 | * \f]
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41 | * by using Householder transformations. Here, \b P is a permutation matrix, \b Q a unitary matrix and \b R an
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42 | * upper triangular matrix.
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43 | *
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44 | * This decomposition performs a very prudent full pivoting in order to be rank-revealing and achieve optimal
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45 | * numerical stability. The trade-off is that it is slower than HouseholderQR and ColPivHouseholderQR.
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46 | *
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47 | * \sa MatrixBase::fullPivHouseholderQr()
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48 | */
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49 | template<typename _MatrixType> class FullPivHouseholderQR
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50 | {
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51 | public:
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52 |
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53 | typedef _MatrixType MatrixType;
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54 | enum {
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55 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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56 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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57 | Options = MatrixType::Options,
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58 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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59 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
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60 | };
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61 | typedef typename MatrixType::Scalar Scalar;
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62 | typedef typename MatrixType::RealScalar RealScalar;
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63 | typedef typename MatrixType::Index Index;
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64 | typedef internal::FullPivHouseholderQRMatrixQReturnType<MatrixType> MatrixQReturnType;
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65 | typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
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66 | typedef Matrix<Index, 1,
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67 | EIGEN_SIZE_MIN_PREFER_DYNAMIC(ColsAtCompileTime,RowsAtCompileTime), RowMajor, 1,
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68 | EIGEN_SIZE_MIN_PREFER_FIXED(MaxColsAtCompileTime,MaxRowsAtCompileTime)> IntDiagSizeVectorType;
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69 | typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationType;
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70 | typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
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71 | typedef typename internal::plain_col_type<MatrixType>::type ColVectorType;
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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 FullPivHouseholderQR::compute(const MatrixType&).
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77 | */
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78 | FullPivHouseholderQR()
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79 | : m_qr(),
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80 | m_hCoeffs(),
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81 | m_rows_transpositions(),
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82 | m_cols_transpositions(),
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83 | m_cols_permutation(),
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84 | m_temp(),
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85 | m_isInitialized(false),
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86 | m_usePrescribedThreshold(false) {}
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87 |
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88 | /** \brief Default Constructor with memory preallocation
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89 | *
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90 | * Like the default constructor but with preallocation of the internal data
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91 | * according to the specified problem \a size.
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92 | * \sa FullPivHouseholderQR()
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93 | */
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94 | FullPivHouseholderQR(Index rows, Index cols)
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95 | : m_qr(rows, cols),
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96 | m_hCoeffs((std::min)(rows,cols)),
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97 | m_rows_transpositions((std::min)(rows,cols)),
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98 | m_cols_transpositions((std::min)(rows,cols)),
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99 | m_cols_permutation(cols),
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100 | m_temp(cols),
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101 | m_isInitialized(false),
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102 | m_usePrescribedThreshold(false) {}
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103 |
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104 | /** \brief Constructs a QR factorization from a given matrix
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105 | *
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106 | * This constructor computes the QR factorization of the matrix \a matrix by calling
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107 | * the method compute(). It is a short cut for:
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108 | *
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109 | * \code
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110 | * FullPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
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111 | * qr.compute(matrix);
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112 | * \endcode
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113 | *
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114 | * \sa compute()
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115 | */
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116 | FullPivHouseholderQR(const MatrixType& matrix)
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117 | : m_qr(matrix.rows(), matrix.cols()),
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118 | m_hCoeffs((std::min)(matrix.rows(), matrix.cols())),
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119 | m_rows_transpositions((std::min)(matrix.rows(), matrix.cols())),
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120 | m_cols_transpositions((std::min)(matrix.rows(), matrix.cols())),
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121 | m_cols_permutation(matrix.cols()),
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122 | m_temp(matrix.cols()),
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123 | m_isInitialized(false),
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124 | m_usePrescribedThreshold(false)
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125 | {
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126 | compute(matrix);
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127 | }
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128 |
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129 | /** This method finds a solution x to the equation Ax=b, where A is the matrix of which
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130 | * \c *this is the QR decomposition.
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131 | *
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132 | * \param b the right-hand-side of the equation to solve.
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133 | *
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134 | * \returns the exact or least-square solution if the rank is greater or equal to the number of columns of A,
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135 | * and an arbitrary solution otherwise.
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136 | *
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137 | * \note The case where b is a matrix is not yet implemented. Also, this
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138 | * code is space inefficient.
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139 | *
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140 | * \note_about_checking_solutions
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141 | *
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142 | * \note_about_arbitrary_choice_of_solution
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143 | *
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144 | * Example: \include FullPivHouseholderQR_solve.cpp
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145 | * Output: \verbinclude FullPivHouseholderQR_solve.out
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146 | */
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147 | template<typename Rhs>
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148 | inline const internal::solve_retval<FullPivHouseholderQR, Rhs>
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149 | solve(const MatrixBase<Rhs>& b) const
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150 | {
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151 | eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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152 | return internal::solve_retval<FullPivHouseholderQR, Rhs>(*this, b.derived());
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153 | }
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154 |
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155 | /** \returns Expression object representing the matrix Q
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156 | */
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157 | MatrixQReturnType matrixQ(void) const;
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158 |
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159 | /** \returns a reference to the matrix where the Householder QR decomposition is stored
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160 | */
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161 | const MatrixType& matrixQR() const
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162 | {
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163 | eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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164 | return m_qr;
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165 | }
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166 |
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167 | FullPivHouseholderQR& compute(const MatrixType& matrix);
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168 |
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169 | /** \returns a const reference to the column permutation matrix */
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170 | const PermutationType& colsPermutation() const
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171 | {
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172 | eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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173 | return m_cols_permutation;
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174 | }
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175 |
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176 | /** \returns a const reference to the vector of indices representing the rows transpositions */
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177 | const IntDiagSizeVectorType& rowsTranspositions() const
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178 | {
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179 | eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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180 | return m_rows_transpositions;
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181 | }
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182 |
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183 | /** \returns the absolute value of the determinant of the matrix of which
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184 | * *this is the QR decomposition. It has only linear complexity
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185 | * (that is, O(n) where n is the dimension of the square matrix)
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186 | * as the QR decomposition has already been computed.
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187 | *
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188 | * \note This is only for square matrices.
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189 | *
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190 | * \warning a determinant can be very big or small, so for matrices
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191 | * of large enough dimension, there is a risk of overflow/underflow.
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192 | * One way to work around that is to use logAbsDeterminant() instead.
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193 | *
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194 | * \sa logAbsDeterminant(), MatrixBase::determinant()
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195 | */
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196 | typename MatrixType::RealScalar absDeterminant() const;
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197 |
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198 | /** \returns the natural log of the absolute value of the determinant of the matrix of which
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199 | * *this is the QR decomposition. It has only linear complexity
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200 | * (that is, O(n) where n is the dimension of the square matrix)
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201 | * as the QR decomposition has already been computed.
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202 | *
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203 | * \note This is only for square matrices.
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204 | *
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205 | * \note This method is useful to work around the risk of overflow/underflow that's inherent
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206 | * to determinant computation.
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207 | *
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208 | * \sa absDeterminant(), MatrixBase::determinant()
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209 | */
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210 | typename MatrixType::RealScalar logAbsDeterminant() const;
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211 |
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212 | /** \returns the rank of the matrix of which *this is the QR decomposition.
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213 | *
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214 | * \note This method has to determine which pivots should be considered nonzero.
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215 | * For that, it uses the threshold value that you can control by calling
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216 | * setThreshold(const RealScalar&).
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217 | */
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218 | inline Index rank() const
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219 | {
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220 | using std::abs;
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221 | eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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222 | RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
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223 | Index result = 0;
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224 | for(Index i = 0; i < m_nonzero_pivots; ++i)
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225 | result += (abs(m_qr.coeff(i,i)) > premultiplied_threshold);
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226 | return result;
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227 | }
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228 |
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229 | /** \returns the dimension of the kernel of the matrix of which *this is the QR decomposition.
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230 | *
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231 | * \note This method has to determine which pivots should be considered nonzero.
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232 | * For that, it uses the threshold value that you can control by calling
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233 | * setThreshold(const RealScalar&).
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234 | */
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235 | inline Index dimensionOfKernel() const
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236 | {
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237 | eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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238 | return cols() - rank();
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239 | }
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240 |
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241 | /** \returns true if the matrix of which *this is the QR decomposition represents an injective
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242 | * linear map, i.e. has trivial kernel; false otherwise.
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243 | *
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244 | * \note This method has to determine which pivots should be considered nonzero.
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245 | * For that, it uses the threshold value that you can control by calling
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246 | * setThreshold(const RealScalar&).
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247 | */
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248 | inline bool isInjective() const
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249 | {
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250 | eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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251 | return rank() == cols();
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252 | }
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253 |
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254 | /** \returns true if the matrix of which *this is the QR decomposition represents a surjective
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255 | * linear map; false otherwise.
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256 | *
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257 | * \note This method has to determine which pivots should be considered nonzero.
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258 | * For that, it uses the threshold value that you can control by calling
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259 | * setThreshold(const RealScalar&).
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260 | */
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261 | inline bool isSurjective() const
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262 | {
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263 | eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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264 | return rank() == rows();
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265 | }
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266 |
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267 | /** \returns true if the matrix of which *this is the QR decomposition is invertible.
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268 | *
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269 | * \note This method has to determine which pivots should be considered nonzero.
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270 | * For that, it uses the threshold value that you can control by calling
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271 | * setThreshold(const RealScalar&).
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272 | */
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273 | inline bool isInvertible() const
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274 | {
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275 | eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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276 | return isInjective() && isSurjective();
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277 | }
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278 |
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279 | /** \returns the inverse of the matrix of which *this is the QR decomposition.
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280 | *
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281 | * \note If this matrix is not invertible, the returned matrix has undefined coefficients.
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282 | * Use isInvertible() to first determine whether this matrix is invertible.
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283 | */ inline const
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284 | internal::solve_retval<FullPivHouseholderQR, typename MatrixType::IdentityReturnType>
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285 | inverse() const
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286 | {
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287 | eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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288 | return internal::solve_retval<FullPivHouseholderQR,typename MatrixType::IdentityReturnType>
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289 | (*this, MatrixType::Identity(m_qr.rows(), m_qr.cols()));
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290 | }
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291 |
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292 | inline Index rows() const { return m_qr.rows(); }
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293 | inline Index cols() const { return m_qr.cols(); }
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294 |
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295 | /** \returns a const reference to the vector of Householder coefficients used to represent the factor \c Q.
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296 | *
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297 | * For advanced uses only.
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298 | */
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299 | const HCoeffsType& hCoeffs() const { return m_hCoeffs; }
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300 |
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301 | /** Allows to prescribe a threshold to be used by certain methods, such as rank(),
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302 | * who need to determine when pivots are to be considered nonzero. This is not used for the
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303 | * QR decomposition itself.
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304 | *
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305 | * When it needs to get the threshold value, Eigen calls threshold(). By default, this
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306 | * uses a formula to automatically determine a reasonable threshold.
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307 | * Once you have called the present method setThreshold(const RealScalar&),
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308 | * your value is used instead.
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309 | *
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310 | * \param threshold The new value to use as the threshold.
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311 | *
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312 | * A pivot will be considered nonzero if its absolute value is strictly greater than
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313 | * \f$ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert \f$
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314 | * where maxpivot is the biggest pivot.
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315 | *
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316 | * If you want to come back to the default behavior, call setThreshold(Default_t)
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317 | */
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318 | FullPivHouseholderQR& setThreshold(const RealScalar& threshold)
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319 | {
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320 | m_usePrescribedThreshold = true;
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321 | m_prescribedThreshold = threshold;
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322 | return *this;
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323 | }
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324 |
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325 | /** Allows to come back to the default behavior, letting Eigen use its default formula for
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326 | * determining the threshold.
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327 | *
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328 | * You should pass the special object Eigen::Default as parameter here.
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329 | * \code qr.setThreshold(Eigen::Default); \endcode
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330 | *
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331 | * See the documentation of setThreshold(const RealScalar&).
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332 | */
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333 | FullPivHouseholderQR& setThreshold(Default_t)
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334 | {
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335 | m_usePrescribedThreshold = false;
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336 | return *this;
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337 | }
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338 |
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339 | /** Returns the threshold that will be used by certain methods such as rank().
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340 | *
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341 | * See the documentation of setThreshold(const RealScalar&).
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342 | */
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343 | RealScalar threshold() const
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344 | {
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345 | eigen_assert(m_isInitialized || m_usePrescribedThreshold);
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346 | return m_usePrescribedThreshold ? m_prescribedThreshold
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347 | // this formula comes from experimenting (see "LU precision tuning" thread on the list)
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348 | // and turns out to be identical to Higham's formula used already in LDLt.
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349 | : NumTraits<Scalar>::epsilon() * RealScalar(m_qr.diagonalSize());
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350 | }
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351 |
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352 | /** \returns the number of nonzero pivots in the QR decomposition.
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353 | * Here nonzero is meant in the exact sense, not in a fuzzy sense.
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354 | * So that notion isn't really intrinsically interesting, but it is
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355 | * still useful when implementing algorithms.
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356 | *
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357 | * \sa rank()
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358 | */
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359 | inline Index nonzeroPivots() const
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360 | {
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361 | eigen_assert(m_isInitialized && "LU is not initialized.");
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362 | return m_nonzero_pivots;
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363 | }
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364 |
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365 | /** \returns the absolute value of the biggest pivot, i.e. the biggest
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366 | * diagonal coefficient of U.
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367 | */
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368 | RealScalar maxPivot() const { return m_maxpivot; }
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369 |
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370 | protected:
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371 |
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372 | static void check_template_parameters()
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373 | {
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374 | EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
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375 | }
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376 |
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377 | MatrixType m_qr;
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378 | HCoeffsType m_hCoeffs;
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379 | IntDiagSizeVectorType m_rows_transpositions;
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380 | IntDiagSizeVectorType m_cols_transpositions;
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381 | PermutationType m_cols_permutation;
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382 | RowVectorType m_temp;
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383 | bool m_isInitialized, m_usePrescribedThreshold;
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384 | RealScalar m_prescribedThreshold, m_maxpivot;
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385 | Index m_nonzero_pivots;
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386 | RealScalar m_precision;
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387 | Index m_det_pq;
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388 | };
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389 |
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390 | template<typename MatrixType>
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391 | typename MatrixType::RealScalar FullPivHouseholderQR<MatrixType>::absDeterminant() const
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392 | {
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393 | using std::abs;
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394 | eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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395 | eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
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396 | return abs(m_qr.diagonal().prod());
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397 | }
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398 |
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399 | template<typename MatrixType>
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400 | typename MatrixType::RealScalar FullPivHouseholderQR<MatrixType>::logAbsDeterminant() const
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401 | {
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402 | eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
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403 | eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
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404 | return m_qr.diagonal().cwiseAbs().array().log().sum();
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405 | }
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406 |
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407 | /** Performs the QR factorization of the given matrix \a matrix. The result of
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408 | * the factorization is stored into \c *this, and a reference to \c *this
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409 | * is returned.
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410 | *
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411 | * \sa class FullPivHouseholderQR, FullPivHouseholderQR(const MatrixType&)
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412 | */
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413 | template<typename MatrixType>
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414 | FullPivHouseholderQR<MatrixType>& FullPivHouseholderQR<MatrixType>::compute(const MatrixType& matrix)
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415 | {
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416 | check_template_parameters();
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417 |
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418 | using std::abs;
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419 | Index rows = matrix.rows();
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420 | Index cols = matrix.cols();
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421 | Index size = (std::min)(rows,cols);
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422 |
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423 | m_qr = matrix;
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424 | m_hCoeffs.resize(size);
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425 |
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426 | m_temp.resize(cols);
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427 |
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428 | m_precision = NumTraits<Scalar>::epsilon() * RealScalar(size);
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429 |
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430 | m_rows_transpositions.resize(size);
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431 | m_cols_transpositions.resize(size);
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432 | Index number_of_transpositions = 0;
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433 |
|
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434 | RealScalar biggest(0);
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435 |
|
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436 | m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
|
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437 | m_maxpivot = RealScalar(0);
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438 |
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439 | for (Index k = 0; k < size; ++k)
|
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440 | {
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441 | Index row_of_biggest_in_corner, col_of_biggest_in_corner;
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442 | RealScalar biggest_in_corner;
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---|
443 |
|
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444 | biggest_in_corner = m_qr.bottomRightCorner(rows-k, cols-k)
|
---|
445 | .cwiseAbs()
|
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446 | .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
|
---|
447 | row_of_biggest_in_corner += k;
|
---|
448 | col_of_biggest_in_corner += k;
|
---|
449 | if(k==0) biggest = biggest_in_corner;
|
---|
450 |
|
---|
451 | // if the corner is negligible, then we have less than full rank, and we can finish early
|
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452 | if(internal::isMuchSmallerThan(biggest_in_corner, biggest, m_precision))
|
---|
453 | {
|
---|
454 | m_nonzero_pivots = k;
|
---|
455 | for(Index i = k; i < size; i++)
|
---|
456 | {
|
---|
457 | m_rows_transpositions.coeffRef(i) = i;
|
---|
458 | m_cols_transpositions.coeffRef(i) = i;
|
---|
459 | m_hCoeffs.coeffRef(i) = Scalar(0);
|
---|
460 | }
|
---|
461 | break;
|
---|
462 | }
|
---|
463 |
|
---|
464 | m_rows_transpositions.coeffRef(k) = row_of_biggest_in_corner;
|
---|
465 | m_cols_transpositions.coeffRef(k) = col_of_biggest_in_corner;
|
---|
466 | if(k != row_of_biggest_in_corner) {
|
---|
467 | m_qr.row(k).tail(cols-k).swap(m_qr.row(row_of_biggest_in_corner).tail(cols-k));
|
---|
468 | ++number_of_transpositions;
|
---|
469 | }
|
---|
470 | if(k != col_of_biggest_in_corner) {
|
---|
471 | m_qr.col(k).swap(m_qr.col(col_of_biggest_in_corner));
|
---|
472 | ++number_of_transpositions;
|
---|
473 | }
|
---|
474 |
|
---|
475 | RealScalar beta;
|
---|
476 | m_qr.col(k).tail(rows-k).makeHouseholderInPlace(m_hCoeffs.coeffRef(k), beta);
|
---|
477 | m_qr.coeffRef(k,k) = beta;
|
---|
478 |
|
---|
479 | // remember the maximum absolute value of diagonal coefficients
|
---|
480 | if(abs(beta) > m_maxpivot) m_maxpivot = abs(beta);
|
---|
481 |
|
---|
482 | m_qr.bottomRightCorner(rows-k, cols-k-1)
|
---|
483 | .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), m_hCoeffs.coeffRef(k), &m_temp.coeffRef(k+1));
|
---|
484 | }
|
---|
485 |
|
---|
486 | m_cols_permutation.setIdentity(cols);
|
---|
487 | for(Index k = 0; k < size; ++k)
|
---|
488 | m_cols_permutation.applyTranspositionOnTheRight(k, m_cols_transpositions.coeff(k));
|
---|
489 |
|
---|
490 | m_det_pq = (number_of_transpositions%2) ? -1 : 1;
|
---|
491 | m_isInitialized = true;
|
---|
492 |
|
---|
493 | return *this;
|
---|
494 | }
|
---|
495 |
|
---|
496 | namespace internal {
|
---|
497 |
|
---|
498 | template<typename _MatrixType, typename Rhs>
|
---|
499 | struct solve_retval<FullPivHouseholderQR<_MatrixType>, Rhs>
|
---|
500 | : solve_retval_base<FullPivHouseholderQR<_MatrixType>, Rhs>
|
---|
501 | {
|
---|
502 | EIGEN_MAKE_SOLVE_HELPERS(FullPivHouseholderQR<_MatrixType>,Rhs)
|
---|
503 |
|
---|
504 | template<typename Dest> void evalTo(Dest& dst) const
|
---|
505 | {
|
---|
506 | const Index rows = dec().rows(), cols = dec().cols();
|
---|
507 | eigen_assert(rhs().rows() == rows);
|
---|
508 |
|
---|
509 | // FIXME introduce nonzeroPivots() and use it here. and more generally,
|
---|
510 | // make the same improvements in this dec as in FullPivLU.
|
---|
511 | if(dec().rank()==0)
|
---|
512 | {
|
---|
513 | dst.setZero();
|
---|
514 | return;
|
---|
515 | }
|
---|
516 |
|
---|
517 | typename Rhs::PlainObject c(rhs());
|
---|
518 |
|
---|
519 | Matrix<Scalar,1,Rhs::ColsAtCompileTime> temp(rhs().cols());
|
---|
520 | for (Index k = 0; k < dec().rank(); ++k)
|
---|
521 | {
|
---|
522 | Index remainingSize = rows-k;
|
---|
523 | c.row(k).swap(c.row(dec().rowsTranspositions().coeff(k)));
|
---|
524 | c.bottomRightCorner(remainingSize, rhs().cols())
|
---|
525 | .applyHouseholderOnTheLeft(dec().matrixQR().col(k).tail(remainingSize-1),
|
---|
526 | dec().hCoeffs().coeff(k), &temp.coeffRef(0));
|
---|
527 | }
|
---|
528 |
|
---|
529 | dec().matrixQR()
|
---|
530 | .topLeftCorner(dec().rank(), dec().rank())
|
---|
531 | .template triangularView<Upper>()
|
---|
532 | .solveInPlace(c.topRows(dec().rank()));
|
---|
533 |
|
---|
534 | for(Index i = 0; i < dec().rank(); ++i) dst.row(dec().colsPermutation().indices().coeff(i)) = c.row(i);
|
---|
535 | for(Index i = dec().rank(); i < cols; ++i) dst.row(dec().colsPermutation().indices().coeff(i)).setZero();
|
---|
536 | }
|
---|
537 | };
|
---|
538 |
|
---|
539 | /** \ingroup QR_Module
|
---|
540 | *
|
---|
541 | * \brief Expression type for return value of FullPivHouseholderQR::matrixQ()
|
---|
542 | *
|
---|
543 | * \tparam MatrixType type of underlying dense matrix
|
---|
544 | */
|
---|
545 | template<typename MatrixType> struct FullPivHouseholderQRMatrixQReturnType
|
---|
546 | : public ReturnByValue<FullPivHouseholderQRMatrixQReturnType<MatrixType> >
|
---|
547 | {
|
---|
548 | public:
|
---|
549 | typedef typename MatrixType::Index Index;
|
---|
550 | typedef typename FullPivHouseholderQR<MatrixType>::IntDiagSizeVectorType IntDiagSizeVectorType;
|
---|
551 | typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
|
---|
552 | typedef Matrix<typename MatrixType::Scalar, 1, MatrixType::RowsAtCompileTime, RowMajor, 1,
|
---|
553 | MatrixType::MaxRowsAtCompileTime> WorkVectorType;
|
---|
554 |
|
---|
555 | FullPivHouseholderQRMatrixQReturnType(const MatrixType& qr,
|
---|
556 | const HCoeffsType& hCoeffs,
|
---|
557 | const IntDiagSizeVectorType& rowsTranspositions)
|
---|
558 | : m_qr(qr),
|
---|
559 | m_hCoeffs(hCoeffs),
|
---|
560 | m_rowsTranspositions(rowsTranspositions)
|
---|
561 | {}
|
---|
562 |
|
---|
563 | template <typename ResultType>
|
---|
564 | void evalTo(ResultType& result) const
|
---|
565 | {
|
---|
566 | const Index rows = m_qr.rows();
|
---|
567 | WorkVectorType workspace(rows);
|
---|
568 | evalTo(result, workspace);
|
---|
569 | }
|
---|
570 |
|
---|
571 | template <typename ResultType>
|
---|
572 | void evalTo(ResultType& result, WorkVectorType& workspace) const
|
---|
573 | {
|
---|
574 | using numext::conj;
|
---|
575 | // compute the product H'_0 H'_1 ... H'_n-1,
|
---|
576 | // where H_k is the k-th Householder transformation I - h_k v_k v_k'
|
---|
577 | // and v_k is the k-th Householder vector [1,m_qr(k+1,k), m_qr(k+2,k), ...]
|
---|
578 | const Index rows = m_qr.rows();
|
---|
579 | const Index cols = m_qr.cols();
|
---|
580 | const Index size = (std::min)(rows, cols);
|
---|
581 | workspace.resize(rows);
|
---|
582 | result.setIdentity(rows, rows);
|
---|
583 | for (Index k = size-1; k >= 0; k--)
|
---|
584 | {
|
---|
585 | result.block(k, k, rows-k, rows-k)
|
---|
586 | .applyHouseholderOnTheLeft(m_qr.col(k).tail(rows-k-1), conj(m_hCoeffs.coeff(k)), &workspace.coeffRef(k));
|
---|
587 | result.row(k).swap(result.row(m_rowsTranspositions.coeff(k)));
|
---|
588 | }
|
---|
589 | }
|
---|
590 |
|
---|
591 | Index rows() const { return m_qr.rows(); }
|
---|
592 | Index cols() const { return m_qr.rows(); }
|
---|
593 |
|
---|
594 | protected:
|
---|
595 | typename MatrixType::Nested m_qr;
|
---|
596 | typename HCoeffsType::Nested m_hCoeffs;
|
---|
597 | typename IntDiagSizeVectorType::Nested m_rowsTranspositions;
|
---|
598 | };
|
---|
599 |
|
---|
600 | } // end namespace internal
|
---|
601 |
|
---|
602 | template<typename MatrixType>
|
---|
603 | inline typename FullPivHouseholderQR<MatrixType>::MatrixQReturnType FullPivHouseholderQR<MatrixType>::matrixQ() const
|
---|
604 | {
|
---|
605 | eigen_assert(m_isInitialized && "FullPivHouseholderQR is not initialized.");
|
---|
606 | return MatrixQReturnType(m_qr, m_hCoeffs, m_rows_transpositions);
|
---|
607 | }
|
---|
608 |
|
---|
609 | /** \return the full-pivoting Householder QR decomposition of \c *this.
|
---|
610 | *
|
---|
611 | * \sa class FullPivHouseholderQR
|
---|
612 | */
|
---|
613 | template<typename Derived>
|
---|
614 | const FullPivHouseholderQR<typename MatrixBase<Derived>::PlainObject>
|
---|
615 | MatrixBase<Derived>::fullPivHouseholderQr() const
|
---|
616 | {
|
---|
617 | return FullPivHouseholderQR<PlainObject>(eval());
|
---|
618 | }
|
---|
619 |
|
---|
620 | } // end namespace Eigen
|
---|
621 |
|
---|
622 | #endif // EIGEN_FULLPIVOTINGHOUSEHOLDERQR_H
|
---|