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) 2008-2010 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 | // Copyright (C) 2010 Vincent Lejeune
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7 | //
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8 | // This Source Code Form is subject to the terms of the Mozilla
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9 | // Public License v. 2.0. If a copy of the MPL was not distributed
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10 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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11 |
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12 | #ifndef EIGEN_QR_H
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13 | #define EIGEN_QR_H
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14 |
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15 | namespace Eigen {
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16 |
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17 | /** \ingroup QR_Module
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18 | *
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19 | *
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20 | * \class HouseholderQR
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21 | *
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22 | * \brief Householder QR decomposition of a matrix
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23 | *
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24 | * \param MatrixType the type of the matrix of which we are computing the QR decomposition
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25 | *
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26 | * This class performs a QR decomposition of a matrix \b A into matrices \b Q and \b R
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27 | * such that
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28 | * \f[
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29 | * \mathbf{A} = \mathbf{Q} \, \mathbf{R}
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30 | * \f]
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31 | * by using Householder transformations. Here, \b Q a unitary matrix and \b R an upper triangular matrix.
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32 | * The result is stored in a compact way compatible with LAPACK.
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33 | *
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34 | * Note that no pivoting is performed. This is \b not a rank-revealing decomposition.
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35 | * If you want that feature, use FullPivHouseholderQR or ColPivHouseholderQR instead.
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36 | *
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37 | * This Householder QR decomposition is faster, but less numerically stable and less feature-full than
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38 | * FullPivHouseholderQR or ColPivHouseholderQR.
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39 | *
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40 | * \sa MatrixBase::householderQr()
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41 | */
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42 | template<typename _MatrixType> class HouseholderQR
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43 | {
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44 | public:
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45 |
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46 | typedef _MatrixType MatrixType;
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47 | enum {
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48 | RowsAtCompileTime = MatrixType::RowsAtCompileTime,
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49 | ColsAtCompileTime = MatrixType::ColsAtCompileTime,
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50 | Options = MatrixType::Options,
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51 | MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
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52 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
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53 | };
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54 | typedef typename MatrixType::Scalar Scalar;
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55 | typedef typename MatrixType::RealScalar RealScalar;
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56 | typedef typename MatrixType::Index Index;
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57 | typedef Matrix<Scalar, RowsAtCompileTime, RowsAtCompileTime, (MatrixType::Flags&RowMajorBit) ? RowMajor : ColMajor, MaxRowsAtCompileTime, MaxRowsAtCompileTime> MatrixQType;
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58 | typedef typename internal::plain_diag_type<MatrixType>::type HCoeffsType;
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59 | typedef typename internal::plain_row_type<MatrixType>::type RowVectorType;
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60 | typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type> HouseholderSequenceType;
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61 |
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62 | /**
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63 | * \brief Default Constructor.
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64 | *
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65 | * The default constructor is useful in cases in which the user intends to
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66 | * perform decompositions via HouseholderQR::compute(const MatrixType&).
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67 | */
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68 | HouseholderQR() : m_qr(), m_hCoeffs(), m_temp(), m_isInitialized(false) {}
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69 |
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70 | /** \brief Default Constructor with memory preallocation
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71 | *
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72 | * Like the default constructor but with preallocation of the internal data
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73 | * according to the specified problem \a size.
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74 | * \sa HouseholderQR()
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75 | */
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76 | HouseholderQR(Index rows, Index cols)
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77 | : m_qr(rows, cols),
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78 | m_hCoeffs((std::min)(rows,cols)),
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79 | m_temp(cols),
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80 | m_isInitialized(false) {}
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81 |
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82 | /** \brief Constructs a QR factorization from a given matrix
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83 | *
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84 | * This constructor computes the QR factorization of the matrix \a matrix by calling
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85 | * the method compute(). It is a short cut for:
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86 | *
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87 | * \code
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88 | * HouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
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89 | * qr.compute(matrix);
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90 | * \endcode
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91 | *
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92 | * \sa compute()
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93 | */
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94 | HouseholderQR(const MatrixType& matrix)
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95 | : m_qr(matrix.rows(), matrix.cols()),
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96 | m_hCoeffs((std::min)(matrix.rows(),matrix.cols())),
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97 | m_temp(matrix.cols()),
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98 | m_isInitialized(false)
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99 | {
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100 | compute(matrix);
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101 | }
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102 |
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103 | /** This method finds a solution x to the equation Ax=b, where A is the matrix of which
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104 | * *this is the QR decomposition, if any exists.
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105 | *
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106 | * \param b the right-hand-side of the equation to solve.
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107 | *
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108 | * \returns a solution.
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109 | *
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110 | * \note The case where b is a matrix is not yet implemented. Also, this
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111 | * code is space inefficient.
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112 | *
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113 | * \note_about_checking_solutions
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114 | *
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115 | * \note_about_arbitrary_choice_of_solution
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116 | *
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117 | * Example: \include HouseholderQR_solve.cpp
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118 | * Output: \verbinclude HouseholderQR_solve.out
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119 | */
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120 | template<typename Rhs>
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121 | inline const internal::solve_retval<HouseholderQR, Rhs>
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122 | solve(const MatrixBase<Rhs>& b) const
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123 | {
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124 | eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
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125 | return internal::solve_retval<HouseholderQR, Rhs>(*this, b.derived());
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126 | }
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127 |
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128 | /** This method returns an expression of the unitary matrix Q as a sequence of Householder transformations.
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129 | *
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130 | * The returned expression can directly be used to perform matrix products. It can also be assigned to a dense Matrix object.
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131 | * Here is an example showing how to recover the full or thin matrix Q, as well as how to perform matrix products using operator*:
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132 | *
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133 | * Example: \include HouseholderQR_householderQ.cpp
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134 | * Output: \verbinclude HouseholderQR_householderQ.out
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135 | */
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136 | HouseholderSequenceType householderQ() const
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137 | {
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138 | eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
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139 | return HouseholderSequenceType(m_qr, m_hCoeffs.conjugate());
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140 | }
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141 |
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142 | /** \returns a reference to the matrix where the Householder QR decomposition is stored
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143 | * in a LAPACK-compatible way.
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144 | */
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145 | const MatrixType& matrixQR() const
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146 | {
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147 | eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
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148 | return m_qr;
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149 | }
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150 |
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151 | HouseholderQR& compute(const MatrixType& matrix);
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152 |
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153 | /** \returns the absolute value of the determinant of the matrix of which
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154 | * *this is the QR decomposition. It has only linear complexity
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155 | * (that is, O(n) where n is the dimension of the square matrix)
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156 | * as the QR decomposition has already been computed.
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157 | *
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158 | * \note This is only for square matrices.
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159 | *
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160 | * \warning a determinant can be very big or small, so for matrices
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161 | * of large enough dimension, there is a risk of overflow/underflow.
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162 | * One way to work around that is to use logAbsDeterminant() instead.
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163 | *
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164 | * \sa logAbsDeterminant(), MatrixBase::determinant()
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165 | */
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166 | typename MatrixType::RealScalar absDeterminant() const;
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167 |
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168 | /** \returns the natural log of the absolute value of the determinant of the matrix of which
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169 | * *this is the QR decomposition. It has only linear complexity
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170 | * (that is, O(n) where n is the dimension of the square matrix)
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171 | * as the QR decomposition has already been computed.
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172 | *
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173 | * \note This is only for square matrices.
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174 | *
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175 | * \note This method is useful to work around the risk of overflow/underflow that's inherent
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176 | * to determinant computation.
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177 | *
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178 | * \sa absDeterminant(), MatrixBase::determinant()
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179 | */
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180 | typename MatrixType::RealScalar logAbsDeterminant() const;
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181 |
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182 | inline Index rows() const { return m_qr.rows(); }
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183 | inline Index cols() const { return m_qr.cols(); }
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184 |
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185 | /** \returns a const reference to the vector of Householder coefficients used to represent the factor \c Q.
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186 | *
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187 | * For advanced uses only.
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188 | */
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189 | const HCoeffsType& hCoeffs() const { return m_hCoeffs; }
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190 |
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191 | protected:
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192 |
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193 | static void check_template_parameters()
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194 | {
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195 | EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
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196 | }
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197 |
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198 | MatrixType m_qr;
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199 | HCoeffsType m_hCoeffs;
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200 | RowVectorType m_temp;
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201 | bool m_isInitialized;
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202 | };
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203 |
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204 | template<typename MatrixType>
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205 | typename MatrixType::RealScalar HouseholderQR<MatrixType>::absDeterminant() const
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206 | {
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207 | using std::abs;
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208 | eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
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209 | eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
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210 | return abs(m_qr.diagonal().prod());
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211 | }
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212 |
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213 | template<typename MatrixType>
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214 | typename MatrixType::RealScalar HouseholderQR<MatrixType>::logAbsDeterminant() const
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215 | {
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216 | eigen_assert(m_isInitialized && "HouseholderQR is not initialized.");
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217 | eigen_assert(m_qr.rows() == m_qr.cols() && "You can't take the determinant of a non-square matrix!");
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218 | return m_qr.diagonal().cwiseAbs().array().log().sum();
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219 | }
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220 |
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221 | namespace internal {
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222 |
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223 | /** \internal */
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224 | template<typename MatrixQR, typename HCoeffs>
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225 | void householder_qr_inplace_unblocked(MatrixQR& mat, HCoeffs& hCoeffs, typename MatrixQR::Scalar* tempData = 0)
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226 | {
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227 | typedef typename MatrixQR::Index Index;
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228 | typedef typename MatrixQR::Scalar Scalar;
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229 | typedef typename MatrixQR::RealScalar RealScalar;
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230 | Index rows = mat.rows();
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231 | Index cols = mat.cols();
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232 | Index size = (std::min)(rows,cols);
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233 |
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234 | eigen_assert(hCoeffs.size() == size);
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235 |
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236 | typedef Matrix<Scalar,MatrixQR::ColsAtCompileTime,1> TempType;
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237 | TempType tempVector;
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238 | if(tempData==0)
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239 | {
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240 | tempVector.resize(cols);
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241 | tempData = tempVector.data();
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242 | }
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243 |
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244 | for(Index k = 0; k < size; ++k)
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245 | {
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246 | Index remainingRows = rows - k;
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247 | Index remainingCols = cols - k - 1;
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248 |
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249 | RealScalar beta;
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250 | mat.col(k).tail(remainingRows).makeHouseholderInPlace(hCoeffs.coeffRef(k), beta);
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251 | mat.coeffRef(k,k) = beta;
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252 |
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253 | // apply H to remaining part of m_qr from the left
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254 | mat.bottomRightCorner(remainingRows, remainingCols)
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255 | .applyHouseholderOnTheLeft(mat.col(k).tail(remainingRows-1), hCoeffs.coeffRef(k), tempData+k+1);
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256 | }
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257 | }
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258 |
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259 | /** \internal */
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260 | template<typename MatrixQR, typename HCoeffs,
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261 | typename MatrixQRScalar = typename MatrixQR::Scalar,
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262 | bool InnerStrideIsOne = (MatrixQR::InnerStrideAtCompileTime == 1 && HCoeffs::InnerStrideAtCompileTime == 1)>
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263 | struct householder_qr_inplace_blocked
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264 | {
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265 | // This is specialized for MKL-supported Scalar types in HouseholderQR_MKL.h
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266 | static void run(MatrixQR& mat, HCoeffs& hCoeffs,
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267 | typename MatrixQR::Index maxBlockSize=32,
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268 | typename MatrixQR::Scalar* tempData = 0)
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269 | {
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270 | typedef typename MatrixQR::Index Index;
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271 | typedef typename MatrixQR::Scalar Scalar;
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272 | typedef Block<MatrixQR,Dynamic,Dynamic> BlockType;
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273 |
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274 | Index rows = mat.rows();
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275 | Index cols = mat.cols();
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276 | Index size = (std::min)(rows, cols);
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277 |
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278 | typedef Matrix<Scalar,Dynamic,1,ColMajor,MatrixQR::MaxColsAtCompileTime,1> TempType;
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279 | TempType tempVector;
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280 | if(tempData==0)
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281 | {
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282 | tempVector.resize(cols);
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283 | tempData = tempVector.data();
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284 | }
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285 |
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286 | Index blockSize = (std::min)(maxBlockSize,size);
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287 |
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288 | Index k = 0;
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289 | for (k = 0; k < size; k += blockSize)
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290 | {
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291 | Index bs = (std::min)(size-k,blockSize); // actual size of the block
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292 | Index tcols = cols - k - bs; // trailing columns
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293 | Index brows = rows-k; // rows of the block
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294 |
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295 | // partition the matrix:
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296 | // A00 | A01 | A02
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297 | // mat = A10 | A11 | A12
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298 | // A20 | A21 | A22
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299 | // and performs the qr dec of [A11^T A12^T]^T
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300 | // and update [A21^T A22^T]^T using level 3 operations.
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301 | // Finally, the algorithm continue on A22
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302 |
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303 | BlockType A11_21 = mat.block(k,k,brows,bs);
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304 | Block<HCoeffs,Dynamic,1> hCoeffsSegment = hCoeffs.segment(k,bs);
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305 |
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306 | householder_qr_inplace_unblocked(A11_21, hCoeffsSegment, tempData);
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307 |
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308 | if(tcols)
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309 | {
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310 | BlockType A21_22 = mat.block(k,k+bs,brows,tcols);
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311 | apply_block_householder_on_the_left(A21_22,A11_21,hCoeffsSegment.adjoint());
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312 | }
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313 | }
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314 | }
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315 | };
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316 |
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317 | template<typename _MatrixType, typename Rhs>
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318 | struct solve_retval<HouseholderQR<_MatrixType>, Rhs>
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319 | : solve_retval_base<HouseholderQR<_MatrixType>, Rhs>
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320 | {
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321 | EIGEN_MAKE_SOLVE_HELPERS(HouseholderQR<_MatrixType>,Rhs)
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322 |
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323 | template<typename Dest> void evalTo(Dest& dst) const
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324 | {
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325 | const Index rows = dec().rows(), cols = dec().cols();
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326 | const Index rank = (std::min)(rows, cols);
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327 | eigen_assert(rhs().rows() == rows);
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328 |
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329 | typename Rhs::PlainObject c(rhs());
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330 |
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331 | // Note that the matrix Q = H_0^* H_1^*... so its inverse is Q^* = (H_0 H_1 ...)^T
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332 | c.applyOnTheLeft(householderSequence(
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333 | dec().matrixQR().leftCols(rank),
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334 | dec().hCoeffs().head(rank)).transpose()
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335 | );
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336 |
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337 | dec().matrixQR()
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338 | .topLeftCorner(rank, rank)
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339 | .template triangularView<Upper>()
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340 | .solveInPlace(c.topRows(rank));
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341 |
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342 | dst.topRows(rank) = c.topRows(rank);
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343 | dst.bottomRows(cols-rank).setZero();
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344 | }
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345 | };
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346 |
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347 | } // end namespace internal
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348 |
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349 | /** Performs the QR factorization of the given matrix \a matrix. The result of
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350 | * the factorization is stored into \c *this, and a reference to \c *this
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351 | * is returned.
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352 | *
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353 | * \sa class HouseholderQR, HouseholderQR(const MatrixType&)
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354 | */
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355 | template<typename MatrixType>
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356 | HouseholderQR<MatrixType>& HouseholderQR<MatrixType>::compute(const MatrixType& matrix)
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357 | {
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358 | check_template_parameters();
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359 |
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360 | Index rows = matrix.rows();
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361 | Index cols = matrix.cols();
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362 | Index size = (std::min)(rows,cols);
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363 |
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364 | m_qr = matrix;
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365 | m_hCoeffs.resize(size);
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366 |
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367 | m_temp.resize(cols);
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368 |
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369 | internal::householder_qr_inplace_blocked<MatrixType, HCoeffsType>::run(m_qr, m_hCoeffs, 48, m_temp.data());
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370 |
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371 | m_isInitialized = true;
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372 | return *this;
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373 | }
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374 |
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375 | /** \return the Householder QR decomposition of \c *this.
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376 | *
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377 | * \sa class HouseholderQR
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378 | */
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379 | template<typename Derived>
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380 | const HouseholderQR<typename MatrixBase<Derived>::PlainObject>
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381 | MatrixBase<Derived>::householderQr() const
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382 | {
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383 | return HouseholderQR<PlainObject>(eval());
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384 | }
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385 |
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386 | } // end namespace Eigen
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387 |
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388 | #endif // EIGEN_QR_H
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