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) 2012 Désiré Nuentsa-Wakam <desire.nuentsa_wakam@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_INCOMPLETE_CHOlESKY_H
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11 | #define EIGEN_INCOMPLETE_CHOlESKY_H
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12 | #include "Eigen/src/IterativeLinearSolvers/IncompleteLUT.h"
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13 | #include <Eigen/OrderingMethods>
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14 | #include <list>
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15 |
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16 | namespace Eigen {
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17 | /**
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18 | * \brief Modified Incomplete Cholesky with dual threshold
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19 | *
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20 | * References : C-J. Lin and J. J. Moré, Incomplete Cholesky Factorizations with
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21 | * Limited memory, SIAM J. Sci. Comput. 21(1), pp. 24-45, 1999
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22 | *
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23 | * \tparam _MatrixType The type of the sparse matrix. It should be a symmetric
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24 | * matrix. It is advised to give a row-oriented sparse matrix
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25 | * \tparam _UpLo The triangular part of the matrix to reference.
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26 | * \tparam _OrderingType
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27 | */
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28 |
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29 | template <typename Scalar, int _UpLo = Lower, typename _OrderingType = NaturalOrdering<int> >
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30 | class IncompleteCholesky : internal::noncopyable
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31 | {
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32 | public:
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33 | typedef SparseMatrix<Scalar,ColMajor> MatrixType;
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34 | typedef _OrderingType OrderingType;
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35 | typedef typename MatrixType::RealScalar RealScalar;
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36 | typedef typename MatrixType::Index Index;
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37 | typedef PermutationMatrix<Dynamic, Dynamic, Index> PermutationType;
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38 | typedef Matrix<Scalar,Dynamic,1> ScalarType;
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39 | typedef Matrix<Index,Dynamic, 1> IndexType;
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40 | typedef std::vector<std::list<Index> > VectorList;
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41 | enum { UpLo = _UpLo };
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42 | public:
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43 | IncompleteCholesky() : m_shift(1),m_factorizationIsOk(false) {}
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44 | IncompleteCholesky(const MatrixType& matrix) : m_shift(1),m_factorizationIsOk(false)
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45 | {
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46 | compute(matrix);
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47 | }
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48 |
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49 | Index rows() const { return m_L.rows(); }
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50 |
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51 | Index cols() const { return m_L.cols(); }
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52 |
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53 |
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54 | /** \brief Reports whether previous computation was successful.
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55 | *
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56 | * \returns \c Success if computation was succesful,
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57 | * \c NumericalIssue if the matrix appears to be negative.
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58 | */
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59 | ComputationInfo info() const
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60 | {
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61 | eigen_assert(m_isInitialized && "IncompleteLLT is not initialized.");
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62 | return m_info;
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63 | }
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64 |
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65 | /**
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66 | * \brief Set the initial shift parameter
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67 | */
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68 | void setShift( Scalar shift) { m_shift = shift; }
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69 |
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70 | /**
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71 | * \brief Computes the fill reducing permutation vector.
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72 | */
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73 | template<typename MatrixType>
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74 | void analyzePattern(const MatrixType& mat)
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75 | {
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76 | OrderingType ord;
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77 | ord(mat.template selfadjointView<UpLo>(), m_perm);
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78 | m_analysisIsOk = true;
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79 | }
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80 |
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81 | template<typename MatrixType>
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82 | void factorize(const MatrixType& amat);
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83 |
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84 | template<typename MatrixType>
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85 | void compute (const MatrixType& matrix)
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86 | {
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87 | analyzePattern(matrix);
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88 | factorize(matrix);
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89 | }
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90 |
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91 | template<typename Rhs, typename Dest>
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92 | void _solve(const Rhs& b, Dest& x) const
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93 | {
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94 | eigen_assert(m_factorizationIsOk && "factorize() should be called first");
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95 | if (m_perm.rows() == b.rows())
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96 | x = m_perm.inverse() * b;
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97 | else
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98 | x = b;
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99 | x = m_scal.asDiagonal() * x;
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100 | x = m_L.template triangularView<UnitLower>().solve(x);
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101 | x = m_L.adjoint().template triangularView<Upper>().solve(x);
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102 | if (m_perm.rows() == b.rows())
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103 | x = m_perm * x;
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104 | x = m_scal.asDiagonal() * x;
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105 | }
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106 | template<typename Rhs> inline const internal::solve_retval<IncompleteCholesky, Rhs>
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107 | solve(const MatrixBase<Rhs>& b) const
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108 | {
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109 | eigen_assert(m_factorizationIsOk && "IncompleteLLT did not succeed");
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110 | eigen_assert(m_isInitialized && "IncompleteLLT is not initialized.");
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111 | eigen_assert(cols()==b.rows()
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112 | && "IncompleteLLT::solve(): invalid number of rows of the right hand side matrix b");
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113 | return internal::solve_retval<IncompleteCholesky, Rhs>(*this, b.derived());
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114 | }
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115 | protected:
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116 | SparseMatrix<Scalar,ColMajor> m_L; // The lower part stored in CSC
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117 | ScalarType m_scal; // The vector for scaling the matrix
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118 | Scalar m_shift; //The initial shift parameter
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119 | bool m_analysisIsOk;
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120 | bool m_factorizationIsOk;
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121 | bool m_isInitialized;
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122 | ComputationInfo m_info;
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123 | PermutationType m_perm;
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124 |
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125 | private:
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126 | template <typename IdxType, typename SclType>
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127 | inline void updateList(const IdxType& colPtr, IdxType& rowIdx, SclType& vals, const Index& col, const Index& jk, IndexType& firstElt, VectorList& listCol);
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128 | };
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129 |
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130 | template<typename Scalar, int _UpLo, typename OrderingType>
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131 | template<typename _MatrixType>
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132 | void IncompleteCholesky<Scalar,_UpLo, OrderingType>::factorize(const _MatrixType& mat)
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133 | {
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134 | using std::sqrt;
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135 | using std::min;
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136 | eigen_assert(m_analysisIsOk && "analyzePattern() should be called first");
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137 |
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138 | // Dropping strategies : Keep only the p largest elements per column, where p is the number of elements in the column of the original matrix. Other strategies will be added
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139 |
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140 | // Apply the fill-reducing permutation computed in analyzePattern()
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141 | if (m_perm.rows() == mat.rows() ) // To detect the null permutation
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142 | m_L.template selfadjointView<Lower>() = mat.template selfadjointView<_UpLo>().twistedBy(m_perm);
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143 | else
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144 | m_L.template selfadjointView<Lower>() = mat.template selfadjointView<_UpLo>();
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145 |
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146 | Index n = m_L.cols();
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147 | Index nnz = m_L.nonZeros();
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148 | Map<ScalarType> vals(m_L.valuePtr(), nnz); //values
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149 | Map<IndexType> rowIdx(m_L.innerIndexPtr(), nnz); //Row indices
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150 | Map<IndexType> colPtr( m_L.outerIndexPtr(), n+1); // Pointer to the beginning of each row
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151 | IndexType firstElt(n-1); // for each j, points to the next entry in vals that will be used in the factorization
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152 | VectorList listCol(n); // listCol(j) is a linked list of columns to update column j
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153 | ScalarType curCol(n); // Store a nonzero values in each column
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154 | IndexType irow(n); // Row indices of nonzero elements in each column
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155 |
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156 |
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157 | // Computes the scaling factors
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158 | m_scal.resize(n);
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159 | for (int j = 0; j < n; j++)
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160 | {
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161 | m_scal(j) = m_L.col(j).norm();
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162 | m_scal(j) = sqrt(m_scal(j));
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163 | }
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164 | // Scale and compute the shift for the matrix
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165 | Scalar mindiag = vals[0];
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166 | for (int j = 0; j < n; j++){
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167 | for (int k = colPtr[j]; k < colPtr[j+1]; k++)
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168 | vals[k] /= (m_scal(j) * m_scal(rowIdx[k]));
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169 | mindiag = (min)(vals[colPtr[j]], mindiag);
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170 | }
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171 |
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172 | if(mindiag < Scalar(0.)) m_shift = m_shift - mindiag;
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173 | // Apply the shift to the diagonal elements of the matrix
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174 | for (int j = 0; j < n; j++)
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175 | vals[colPtr[j]] += m_shift;
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176 | // jki version of the Cholesky factorization
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177 | for (int j=0; j < n; ++j)
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178 | {
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179 | //Left-looking factorize the column j
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180 | // First, load the jth column into curCol
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181 | Scalar diag = vals[colPtr[j]]; // It is assumed that only the lower part is stored
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182 | curCol.setZero();
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183 | irow.setLinSpaced(n,0,n-1);
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184 | for (int i = colPtr[j] + 1; i < colPtr[j+1]; i++)
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185 | {
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186 | curCol(rowIdx[i]) = vals[i];
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187 | irow(rowIdx[i]) = rowIdx[i];
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188 | }
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189 | std::list<int>::iterator k;
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190 | // Browse all previous columns that will update column j
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191 | for(k = listCol[j].begin(); k != listCol[j].end(); k++)
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192 | {
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193 | int jk = firstElt(*k); // First element to use in the column
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194 | jk += 1;
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195 | for (int i = jk; i < colPtr[*k+1]; i++)
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196 | {
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197 | curCol(rowIdx[i]) -= vals[i] * vals[jk] ;
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198 | }
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199 | updateList(colPtr,rowIdx,vals, *k, jk, firstElt, listCol);
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200 | }
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201 |
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202 | // Scale the current column
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203 | if(RealScalar(diag) <= 0)
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204 | {
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205 | std::cerr << "\nNegative diagonal during Incomplete factorization... "<< j << "\n";
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206 | m_info = NumericalIssue;
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207 | return;
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208 | }
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209 | RealScalar rdiag = sqrt(RealScalar(diag));
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210 | vals[colPtr[j]] = rdiag;
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211 | for (int i = j+1; i < n; i++)
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212 | {
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213 | //Scale
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214 | curCol(i) /= rdiag;
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215 | //Update the remaining diagonals with curCol
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216 | vals[colPtr[i]] -= curCol(i) * curCol(i);
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217 | }
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218 | // Select the largest p elements
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219 | // p is the original number of elements in the column (without the diagonal)
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220 | int p = colPtr[j+1] - colPtr[j] - 1 ;
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221 | internal::QuickSplit(curCol, irow, p);
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222 | // Insert the largest p elements in the matrix
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223 | int cpt = 0;
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224 | for (int i = colPtr[j]+1; i < colPtr[j+1]; i++)
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225 | {
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226 | vals[i] = curCol(cpt);
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227 | rowIdx[i] = irow(cpt);
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228 | cpt ++;
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229 | }
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230 | // Get the first smallest row index and put it after the diagonal element
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231 | Index jk = colPtr(j)+1;
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232 | updateList(colPtr,rowIdx,vals,j,jk,firstElt,listCol);
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233 | }
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234 | m_factorizationIsOk = true;
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235 | m_isInitialized = true;
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236 | m_info = Success;
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237 | }
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238 |
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239 | template<typename Scalar, int _UpLo, typename OrderingType>
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240 | template <typename IdxType, typename SclType>
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241 | inline void IncompleteCholesky<Scalar,_UpLo, OrderingType>::updateList(const IdxType& colPtr, IdxType& rowIdx, SclType& vals, const Index& col, const Index& jk, IndexType& firstElt, VectorList& listCol)
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242 | {
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243 | if (jk < colPtr(col+1) )
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244 | {
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245 | Index p = colPtr(col+1) - jk;
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246 | Index minpos;
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247 | rowIdx.segment(jk,p).minCoeff(&minpos);
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248 | minpos += jk;
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249 | if (rowIdx(minpos) != rowIdx(jk))
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250 | {
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251 | //Swap
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252 | std::swap(rowIdx(jk),rowIdx(minpos));
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253 | std::swap(vals(jk),vals(minpos));
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254 | }
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255 | firstElt(col) = jk;
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256 | listCol[rowIdx(jk)].push_back(col);
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257 | }
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258 | }
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259 | namespace internal {
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260 |
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261 | template<typename _Scalar, int _UpLo, typename OrderingType, typename Rhs>
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262 | struct solve_retval<IncompleteCholesky<_Scalar, _UpLo, OrderingType>, Rhs>
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263 | : solve_retval_base<IncompleteCholesky<_Scalar, _UpLo, OrderingType>, Rhs>
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264 | {
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265 | typedef IncompleteCholesky<_Scalar, _UpLo, OrderingType> Dec;
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266 | EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs)
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267 |
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268 | template<typename Dest> void evalTo(Dest& dst) const
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269 | {
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270 | dec()._solve(rhs(),dst);
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271 | }
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272 | };
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273 |
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274 | } // end namespace internal
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275 |
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276 | } // end namespace Eigen
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277 |
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278 | #endif
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