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 Gael Guennebaud <gael.guennebaud@inria.fr>
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5 | //
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6 | // This Source Code Form is subject to the terms of the Mozilla
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7 | // Public License v. 2.0. If a copy of the MPL was not distributed
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8 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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9 |
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10 | #ifndef EIGEN_LLT_H
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11 | #define EIGEN_LLT_H
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12 |
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13 | namespace Eigen {
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14 |
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15 | namespace internal{
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16 | template<typename MatrixType, int UpLo> struct LLT_Traits;
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17 | }
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18 |
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19 | /** \ingroup Cholesky_Module
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20 | *
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21 | * \class LLT
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22 | *
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23 | * \brief Standard Cholesky decomposition (LL^T) of a matrix and associated features
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24 | *
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25 | * \param MatrixType the type of the matrix of which we are computing the LL^T Cholesky decomposition
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26 | * \param UpLo the triangular part that will be used for the decompositon: Lower (default) or Upper.
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27 | * The other triangular part won't be read.
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28 | *
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29 | * This class performs a LL^T Cholesky decomposition of a symmetric, positive definite
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30 | * matrix A such that A = LL^* = U^*U, where L is lower triangular.
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31 | *
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32 | * While the Cholesky decomposition is particularly useful to solve selfadjoint problems like D^*D x = b,
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33 | * for that purpose, we recommend the Cholesky decomposition without square root which is more stable
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34 | * and even faster. Nevertheless, this standard Cholesky decomposition remains useful in many other
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35 | * situations like generalised eigen problems with hermitian matrices.
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36 | *
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37 | * Remember that Cholesky decompositions are not rank-revealing. This LLT decomposition is only stable on positive definite matrices,
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38 | * use LDLT instead for the semidefinite case. Also, do not use a Cholesky decomposition to determine whether a system of equations
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39 | * has a solution.
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40 | *
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41 | * Example: \include LLT_example.cpp
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42 | * Output: \verbinclude LLT_example.out
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43 | *
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44 | * \sa MatrixBase::llt(), class LDLT
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45 | */
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46 | /* HEY THIS DOX IS DISABLED BECAUSE THERE's A BUG EITHER HERE OR IN LDLT ABOUT THAT (OR BOTH)
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47 | * Note that during the decomposition, only the upper triangular part of A is considered. Therefore,
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48 | * the strict lower part does not have to store correct values.
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49 | */
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50 | template<typename _MatrixType, int _UpLo> class LLT
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51 | {
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52 | public:
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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 | MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
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59 | };
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60 | typedef typename MatrixType::Scalar Scalar;
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61 | typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
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62 | typedef typename MatrixType::Index Index;
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63 |
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64 | enum {
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65 | PacketSize = internal::packet_traits<Scalar>::size,
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66 | AlignmentMask = int(PacketSize)-1,
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67 | UpLo = _UpLo
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68 | };
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69 |
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70 | typedef internal::LLT_Traits<MatrixType,UpLo> Traits;
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71 |
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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 LLT::compute(const MatrixType&).
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77 | */
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78 | LLT() : m_matrix(), m_isInitialized(false) {}
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79 |
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80 | /** \brief Default Constructor with memory preallocation
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81 | *
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82 | * Like the default constructor but with preallocation of the internal data
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83 | * according to the specified problem \a size.
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84 | * \sa LLT()
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85 | */
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86 | LLT(Index size) : m_matrix(size, size),
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87 | m_isInitialized(false) {}
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88 |
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89 | LLT(const MatrixType& matrix)
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90 | : m_matrix(matrix.rows(), matrix.cols()),
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91 | m_isInitialized(false)
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92 | {
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93 | compute(matrix);
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94 | }
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95 |
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96 | /** \returns a view of the upper triangular matrix U */
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97 | inline typename Traits::MatrixU matrixU() const
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98 | {
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99 | eigen_assert(m_isInitialized && "LLT is not initialized.");
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100 | return Traits::getU(m_matrix);
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101 | }
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102 |
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103 | /** \returns a view of the lower triangular matrix L */
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104 | inline typename Traits::MatrixL matrixL() const
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105 | {
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106 | eigen_assert(m_isInitialized && "LLT is not initialized.");
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107 | return Traits::getL(m_matrix);
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108 | }
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109 |
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110 | /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A.
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111 | *
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112 | * Since this LLT class assumes anyway that the matrix A is invertible, the solution
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113 | * theoretically exists and is unique regardless of b.
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114 | *
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115 | * Example: \include LLT_solve.cpp
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116 | * Output: \verbinclude LLT_solve.out
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117 | *
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118 | * \sa solveInPlace(), MatrixBase::llt()
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119 | */
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120 | template<typename Rhs>
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121 | inline const internal::solve_retval<LLT, 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 && "LLT is not initialized.");
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125 | eigen_assert(m_matrix.rows()==b.rows()
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126 | && "LLT::solve(): invalid number of rows of the right hand side matrix b");
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127 | return internal::solve_retval<LLT, Rhs>(*this, b.derived());
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128 | }
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129 |
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130 | #ifdef EIGEN2_SUPPORT
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131 | template<typename OtherDerived, typename ResultType>
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132 | bool solve(const MatrixBase<OtherDerived>& b, ResultType *result) const
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133 | {
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134 | *result = this->solve(b);
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135 | return true;
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136 | }
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137 |
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138 | bool isPositiveDefinite() const { return true; }
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139 | #endif
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140 |
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141 | template<typename Derived>
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142 | void solveInPlace(MatrixBase<Derived> &bAndX) const;
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143 |
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144 | LLT& compute(const MatrixType& matrix);
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145 |
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146 | /** \returns the LLT decomposition matrix
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147 | *
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148 | * TODO: document the storage layout
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149 | */
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150 | inline const MatrixType& matrixLLT() const
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151 | {
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152 | eigen_assert(m_isInitialized && "LLT is not initialized.");
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153 | return m_matrix;
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154 | }
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155 |
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156 | MatrixType reconstructedMatrix() const;
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157 |
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158 |
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159 | /** \brief Reports whether previous computation was successful.
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160 | *
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161 | * \returns \c Success if computation was succesful,
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162 | * \c NumericalIssue if the matrix.appears to be negative.
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163 | */
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164 | ComputationInfo info() const
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165 | {
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166 | eigen_assert(m_isInitialized && "LLT is not initialized.");
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167 | return m_info;
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168 | }
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169 |
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170 | inline Index rows() const { return m_matrix.rows(); }
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171 | inline Index cols() const { return m_matrix.cols(); }
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172 |
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173 | template<typename VectorType>
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174 | LLT rankUpdate(const VectorType& vec, const RealScalar& sigma = 1);
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175 |
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176 | protected:
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177 |
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178 | static void check_template_parameters()
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179 | {
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180 | EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
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181 | }
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182 |
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183 | /** \internal
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184 | * Used to compute and store L
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185 | * The strict upper part is not used and even not initialized.
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186 | */
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187 | MatrixType m_matrix;
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188 | bool m_isInitialized;
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189 | ComputationInfo m_info;
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190 | };
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191 |
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192 | namespace internal {
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193 |
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194 | template<typename Scalar, int UpLo> struct llt_inplace;
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195 |
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196 | template<typename MatrixType, typename VectorType>
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197 | static typename MatrixType::Index llt_rank_update_lower(MatrixType& mat, const VectorType& vec, const typename MatrixType::RealScalar& sigma)
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198 | {
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199 | using std::sqrt;
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200 | typedef typename MatrixType::Scalar Scalar;
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201 | typedef typename MatrixType::RealScalar RealScalar;
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202 | typedef typename MatrixType::Index Index;
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203 | typedef typename MatrixType::ColXpr ColXpr;
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204 | typedef typename internal::remove_all<ColXpr>::type ColXprCleaned;
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205 | typedef typename ColXprCleaned::SegmentReturnType ColXprSegment;
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206 | typedef Matrix<Scalar,Dynamic,1> TempVectorType;
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207 | typedef typename TempVectorType::SegmentReturnType TempVecSegment;
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208 |
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209 | Index n = mat.cols();
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210 | eigen_assert(mat.rows()==n && vec.size()==n);
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211 |
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212 | TempVectorType temp;
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213 |
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214 | if(sigma>0)
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215 | {
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216 | // This version is based on Givens rotations.
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217 | // It is faster than the other one below, but only works for updates,
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218 | // i.e., for sigma > 0
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219 | temp = sqrt(sigma) * vec;
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220 |
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221 | for(Index i=0; i<n; ++i)
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222 | {
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223 | JacobiRotation<Scalar> g;
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224 | g.makeGivens(mat(i,i), -temp(i), &mat(i,i));
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225 |
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226 | Index rs = n-i-1;
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227 | if(rs>0)
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228 | {
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229 | ColXprSegment x(mat.col(i).tail(rs));
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230 | TempVecSegment y(temp.tail(rs));
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231 | apply_rotation_in_the_plane(x, y, g);
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232 | }
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233 | }
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234 | }
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235 | else
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236 | {
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237 | temp = vec;
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238 | RealScalar beta = 1;
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239 | for(Index j=0; j<n; ++j)
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240 | {
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241 | RealScalar Ljj = numext::real(mat.coeff(j,j));
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242 | RealScalar dj = numext::abs2(Ljj);
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243 | Scalar wj = temp.coeff(j);
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244 | RealScalar swj2 = sigma*numext::abs2(wj);
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245 | RealScalar gamma = dj*beta + swj2;
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246 |
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247 | RealScalar x = dj + swj2/beta;
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248 | if (x<=RealScalar(0))
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249 | return j;
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250 | RealScalar nLjj = sqrt(x);
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251 | mat.coeffRef(j,j) = nLjj;
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252 | beta += swj2/dj;
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253 |
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254 | // Update the terms of L
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255 | Index rs = n-j-1;
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256 | if(rs)
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257 | {
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258 | temp.tail(rs) -= (wj/Ljj) * mat.col(j).tail(rs);
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259 | if(gamma != 0)
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260 | mat.col(j).tail(rs) = (nLjj/Ljj) * mat.col(j).tail(rs) + (nLjj * sigma*numext::conj(wj)/gamma)*temp.tail(rs);
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261 | }
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262 | }
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263 | }
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264 | return -1;
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265 | }
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266 |
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267 | template<typename Scalar> struct llt_inplace<Scalar, Lower>
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268 | {
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269 | typedef typename NumTraits<Scalar>::Real RealScalar;
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270 | template<typename MatrixType>
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271 | static typename MatrixType::Index unblocked(MatrixType& mat)
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272 | {
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273 | using std::sqrt;
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274 | typedef typename MatrixType::Index Index;
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275 |
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276 | eigen_assert(mat.rows()==mat.cols());
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277 | const Index size = mat.rows();
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278 | for(Index k = 0; k < size; ++k)
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279 | {
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280 | Index rs = size-k-1; // remaining size
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281 |
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282 | Block<MatrixType,Dynamic,1> A21(mat,k+1,k,rs,1);
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283 | Block<MatrixType,1,Dynamic> A10(mat,k,0,1,k);
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284 | Block<MatrixType,Dynamic,Dynamic> A20(mat,k+1,0,rs,k);
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285 |
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286 | RealScalar x = numext::real(mat.coeff(k,k));
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287 | if (k>0) x -= A10.squaredNorm();
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288 | if (x<=RealScalar(0))
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289 | return k;
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290 | mat.coeffRef(k,k) = x = sqrt(x);
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291 | if (k>0 && rs>0) A21.noalias() -= A20 * A10.adjoint();
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292 | if (rs>0) A21 /= x;
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293 | }
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294 | return -1;
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295 | }
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296 |
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297 | template<typename MatrixType>
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298 | static typename MatrixType::Index blocked(MatrixType& m)
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299 | {
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300 | typedef typename MatrixType::Index Index;
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301 | eigen_assert(m.rows()==m.cols());
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302 | Index size = m.rows();
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303 | if(size<32)
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304 | return unblocked(m);
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305 |
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306 | Index blockSize = size/8;
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307 | blockSize = (blockSize/16)*16;
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308 | blockSize = (std::min)((std::max)(blockSize,Index(8)), Index(128));
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309 |
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310 | for (Index k=0; k<size; k+=blockSize)
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311 | {
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312 | // partition the matrix:
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313 | // A00 | - | -
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314 | // lu = A10 | A11 | -
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315 | // A20 | A21 | A22
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316 | Index bs = (std::min)(blockSize, size-k);
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317 | Index rs = size - k - bs;
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318 | Block<MatrixType,Dynamic,Dynamic> A11(m,k, k, bs,bs);
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319 | Block<MatrixType,Dynamic,Dynamic> A21(m,k+bs,k, rs,bs);
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320 | Block<MatrixType,Dynamic,Dynamic> A22(m,k+bs,k+bs,rs,rs);
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321 |
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322 | Index ret;
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323 | if((ret=unblocked(A11))>=0) return k+ret;
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324 | if(rs>0) A11.adjoint().template triangularView<Upper>().template solveInPlace<OnTheRight>(A21);
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325 | if(rs>0) A22.template selfadjointView<Lower>().rankUpdate(A21,-1); // bottleneck
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326 | }
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327 | return -1;
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328 | }
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329 |
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330 | template<typename MatrixType, typename VectorType>
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331 | static typename MatrixType::Index rankUpdate(MatrixType& mat, const VectorType& vec, const RealScalar& sigma)
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332 | {
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333 | return Eigen::internal::llt_rank_update_lower(mat, vec, sigma);
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334 | }
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335 | };
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336 |
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337 | template<typename Scalar> struct llt_inplace<Scalar, Upper>
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338 | {
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339 | typedef typename NumTraits<Scalar>::Real RealScalar;
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340 |
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341 | template<typename MatrixType>
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342 | static EIGEN_STRONG_INLINE typename MatrixType::Index unblocked(MatrixType& mat)
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343 | {
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344 | Transpose<MatrixType> matt(mat);
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345 | return llt_inplace<Scalar, Lower>::unblocked(matt);
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346 | }
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347 | template<typename MatrixType>
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348 | static EIGEN_STRONG_INLINE typename MatrixType::Index blocked(MatrixType& mat)
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349 | {
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350 | Transpose<MatrixType> matt(mat);
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351 | return llt_inplace<Scalar, Lower>::blocked(matt);
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352 | }
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353 | template<typename MatrixType, typename VectorType>
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354 | static typename MatrixType::Index rankUpdate(MatrixType& mat, const VectorType& vec, const RealScalar& sigma)
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355 | {
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356 | Transpose<MatrixType> matt(mat);
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357 | return llt_inplace<Scalar, Lower>::rankUpdate(matt, vec.conjugate(), sigma);
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358 | }
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359 | };
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360 |
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361 | template<typename MatrixType> struct LLT_Traits<MatrixType,Lower>
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362 | {
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363 | typedef const TriangularView<const MatrixType, Lower> MatrixL;
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364 | typedef const TriangularView<const typename MatrixType::AdjointReturnType, Upper> MatrixU;
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365 | static inline MatrixL getL(const MatrixType& m) { return m; }
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366 | static inline MatrixU getU(const MatrixType& m) { return m.adjoint(); }
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367 | static bool inplace_decomposition(MatrixType& m)
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368 | { return llt_inplace<typename MatrixType::Scalar, Lower>::blocked(m)==-1; }
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369 | };
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370 |
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371 | template<typename MatrixType> struct LLT_Traits<MatrixType,Upper>
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372 | {
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373 | typedef const TriangularView<const typename MatrixType::AdjointReturnType, Lower> MatrixL;
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374 | typedef const TriangularView<const MatrixType, Upper> MatrixU;
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375 | static inline MatrixL getL(const MatrixType& m) { return m.adjoint(); }
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376 | static inline MatrixU getU(const MatrixType& m) { return m; }
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377 | static bool inplace_decomposition(MatrixType& m)
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378 | { return llt_inplace<typename MatrixType::Scalar, Upper>::blocked(m)==-1; }
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379 | };
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380 |
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381 | } // end namespace internal
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382 |
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383 | /** Computes / recomputes the Cholesky decomposition A = LL^* = U^*U of \a matrix
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384 | *
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385 | * \returns a reference to *this
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386 | *
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387 | * Example: \include TutorialLinAlgComputeTwice.cpp
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388 | * Output: \verbinclude TutorialLinAlgComputeTwice.out
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389 | */
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390 | template<typename MatrixType, int _UpLo>
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391 | LLT<MatrixType,_UpLo>& LLT<MatrixType,_UpLo>::compute(const MatrixType& a)
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392 | {
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393 | check_template_parameters();
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394 |
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395 | eigen_assert(a.rows()==a.cols());
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396 | const Index size = a.rows();
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397 | m_matrix.resize(size, size);
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398 | m_matrix = a;
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399 |
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400 | m_isInitialized = true;
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401 | bool ok = Traits::inplace_decomposition(m_matrix);
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402 | m_info = ok ? Success : NumericalIssue;
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403 |
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404 | return *this;
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405 | }
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406 |
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407 | /** Performs a rank one update (or dowdate) of the current decomposition.
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408 | * If A = LL^* before the rank one update,
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409 | * then after it we have LL^* = A + sigma * v v^* where \a v must be a vector
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410 | * of same dimension.
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411 | */
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412 | template<typename _MatrixType, int _UpLo>
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413 | template<typename VectorType>
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414 | LLT<_MatrixType,_UpLo> LLT<_MatrixType,_UpLo>::rankUpdate(const VectorType& v, const RealScalar& sigma)
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415 | {
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416 | EIGEN_STATIC_ASSERT_VECTOR_ONLY(VectorType);
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417 | eigen_assert(v.size()==m_matrix.cols());
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418 | eigen_assert(m_isInitialized);
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419 | if(internal::llt_inplace<typename MatrixType::Scalar, UpLo>::rankUpdate(m_matrix,v,sigma)>=0)
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420 | m_info = NumericalIssue;
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421 | else
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422 | m_info = Success;
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423 |
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424 | return *this;
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425 | }
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426 |
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427 | namespace internal {
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428 | template<typename _MatrixType, int UpLo, typename Rhs>
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429 | struct solve_retval<LLT<_MatrixType, UpLo>, Rhs>
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430 | : solve_retval_base<LLT<_MatrixType, UpLo>, Rhs>
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431 | {
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432 | typedef LLT<_MatrixType,UpLo> LLTType;
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433 | EIGEN_MAKE_SOLVE_HELPERS(LLTType,Rhs)
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434 |
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435 | template<typename Dest> void evalTo(Dest& dst) const
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436 | {
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437 | dst = rhs();
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438 | dec().solveInPlace(dst);
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439 | }
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440 | };
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441 | }
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442 |
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443 | /** \internal use x = llt_object.solve(x);
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444 | *
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445 | * This is the \em in-place version of solve().
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446 | *
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447 | * \param bAndX represents both the right-hand side matrix b and result x.
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448 | *
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449 | * \returns true always! If you need to check for existence of solutions, use another decomposition like LU, QR, or SVD.
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450 | *
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451 | * This version avoids a copy when the right hand side matrix b is not
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452 | * needed anymore.
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453 | *
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454 | * \sa LLT::solve(), MatrixBase::llt()
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455 | */
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456 | template<typename MatrixType, int _UpLo>
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457 | template<typename Derived>
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458 | void LLT<MatrixType,_UpLo>::solveInPlace(MatrixBase<Derived> &bAndX) const
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459 | {
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460 | eigen_assert(m_isInitialized && "LLT is not initialized.");
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461 | eigen_assert(m_matrix.rows()==bAndX.rows());
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462 | matrixL().solveInPlace(bAndX);
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463 | matrixU().solveInPlace(bAndX);
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464 | }
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465 |
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466 | /** \returns the matrix represented by the decomposition,
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467 | * i.e., it returns the product: L L^*.
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468 | * This function is provided for debug purpose. */
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469 | template<typename MatrixType, int _UpLo>
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470 | MatrixType LLT<MatrixType,_UpLo>::reconstructedMatrix() const
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471 | {
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472 | eigen_assert(m_isInitialized && "LLT is not initialized.");
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473 | return matrixL() * matrixL().adjoint().toDenseMatrix();
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474 | }
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475 |
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476 | /** \cholesky_module
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477 | * \returns the LLT decomposition of \c *this
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478 | */
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479 | template<typename Derived>
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480 | inline const LLT<typename MatrixBase<Derived>::PlainObject>
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481 | MatrixBase<Derived>::llt() const
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482 | {
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483 | return LLT<PlainObject>(derived());
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484 | }
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485 |
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486 | /** \cholesky_module
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487 | * \returns the LLT decomposition of \c *this
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488 | */
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489 | template<typename MatrixType, unsigned int UpLo>
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490 | inline const LLT<typename SelfAdjointView<MatrixType, UpLo>::PlainObject, UpLo>
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491 | SelfAdjointView<MatrixType, UpLo>::llt() const
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492 | {
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493 | return LLT<PlainObject,UpLo>(m_matrix);
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494 | }
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495 |
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496 | } // end namespace Eigen
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497 |
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498 | #endif // EIGEN_LLT_H
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