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_DGMRES_H
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11 | #define EIGEN_DGMRES_H
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12 |
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13 | #include <Eigen/Eigenvalues>
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14 |
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15 | namespace Eigen {
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16 |
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17 | template< typename _MatrixType,
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18 | typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
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19 | class DGMRES;
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20 |
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21 | namespace internal {
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22 |
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23 | template< typename _MatrixType, typename _Preconditioner>
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24 | struct traits<DGMRES<_MatrixType,_Preconditioner> >
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25 | {
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26 | typedef _MatrixType MatrixType;
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27 | typedef _Preconditioner Preconditioner;
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28 | };
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29 |
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30 | /** \brief Computes a permutation vector to have a sorted sequence
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31 | * \param vec The vector to reorder.
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32 | * \param perm gives the sorted sequence on output. Must be initialized with 0..n-1
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33 | * \param ncut Put the ncut smallest elements at the end of the vector
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34 | * WARNING This is an expensive sort, so should be used only
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35 | * for small size vectors
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36 | * TODO Use modified QuickSplit or std::nth_element to get the smallest values
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37 | */
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38 | template <typename VectorType, typename IndexType>
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39 | void sortWithPermutation (VectorType& vec, IndexType& perm, typename IndexType::Scalar& ncut)
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40 | {
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41 | eigen_assert(vec.size() == perm.size());
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42 | typedef typename IndexType::Scalar Index;
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43 | typedef typename VectorType::Scalar Scalar;
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44 | bool flag;
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45 | for (Index k = 0; k < ncut; k++)
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46 | {
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47 | flag = false;
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48 | for (Index j = 0; j < vec.size()-1; j++)
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49 | {
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50 | if ( vec(perm(j)) < vec(perm(j+1)) )
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51 | {
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52 | std::swap(perm(j),perm(j+1));
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53 | flag = true;
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54 | }
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55 | if (!flag) break; // The vector is in sorted order
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56 | }
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57 | }
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58 | }
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59 |
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60 | }
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61 | /**
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62 | * \ingroup IterativeLInearSolvers_Module
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63 | * \brief A Restarted GMRES with deflation.
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64 | * This class implements a modification of the GMRES solver for
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65 | * sparse linear systems. The basis is built with modified
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66 | * Gram-Schmidt. At each restart, a few approximated eigenvectors
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67 | * corresponding to the smallest eigenvalues are used to build a
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68 | * preconditioner for the next cycle. This preconditioner
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69 | * for deflation can be combined with any other preconditioner,
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70 | * the IncompleteLUT for instance. The preconditioner is applied
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71 | * at right of the matrix and the combination is multiplicative.
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72 | *
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73 | * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix.
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74 | * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
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75 | * Typical usage :
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76 | * \code
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77 | * SparseMatrix<double> A;
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78 | * VectorXd x, b;
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79 | * //Fill A and b ...
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80 | * DGMRES<SparseMatrix<double> > solver;
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81 | * solver.set_restart(30); // Set restarting value
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82 | * solver.setEigenv(1); // Set the number of eigenvalues to deflate
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83 | * solver.compute(A);
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84 | * x = solver.solve(b);
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85 | * \endcode
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86 | *
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87 | * References :
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88 | * [1] D. NUENTSA WAKAM and F. PACULL, Memory Efficient Hybrid
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89 | * Algebraic Solvers for Linear Systems Arising from Compressible
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90 | * Flows, Computers and Fluids, In Press,
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91 | * http://dx.doi.org/10.1016/j.compfluid.2012.03.023
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92 | * [2] K. Burrage and J. Erhel, On the performance of various
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93 | * adaptive preconditioned GMRES strategies, 5(1998), 101-121.
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94 | * [3] J. Erhel, K. Burrage and B. Pohl, Restarted GMRES
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95 | * preconditioned by deflation,J. Computational and Applied
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96 | * Mathematics, 69(1996), 303-318.
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97 |
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98 | *
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99 | */
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100 | template< typename _MatrixType, typename _Preconditioner>
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101 | class DGMRES : public IterativeSolverBase<DGMRES<_MatrixType,_Preconditioner> >
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102 | {
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103 | typedef IterativeSolverBase<DGMRES> Base;
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104 | using Base::mp_matrix;
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105 | using Base::m_error;
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106 | using Base::m_iterations;
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107 | using Base::m_info;
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108 | using Base::m_isInitialized;
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109 | using Base::m_tolerance;
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110 | public:
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111 | typedef _MatrixType MatrixType;
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112 | typedef typename MatrixType::Scalar Scalar;
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113 | typedef typename MatrixType::Index Index;
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114 | typedef typename MatrixType::RealScalar RealScalar;
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115 | typedef _Preconditioner Preconditioner;
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116 | typedef Matrix<Scalar,Dynamic,Dynamic> DenseMatrix;
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117 | typedef Matrix<RealScalar,Dynamic,Dynamic> DenseRealMatrix;
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118 | typedef Matrix<Scalar,Dynamic,1> DenseVector;
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119 | typedef Matrix<RealScalar,Dynamic,1> DenseRealVector;
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120 | typedef Matrix<std::complex<RealScalar>, Dynamic, 1> ComplexVector;
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121 |
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122 |
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123 | /** Default constructor. */
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124 | DGMRES() : Base(),m_restart(30),m_neig(0),m_r(0),m_maxNeig(5),m_isDeflAllocated(false),m_isDeflInitialized(false) {}
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125 |
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126 | /** Initialize the solver with matrix \a A for further \c Ax=b solving.
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127 | *
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128 | * This constructor is a shortcut for the default constructor followed
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129 | * by a call to compute().
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130 | *
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131 | * \warning this class stores a reference to the matrix A as well as some
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132 | * precomputed values that depend on it. Therefore, if \a A is changed
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133 | * this class becomes invalid. Call compute() to update it with the new
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134 | * matrix A, or modify a copy of A.
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135 | */
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136 | template<typename MatrixDerived>
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137 | explicit DGMRES(const EigenBase<MatrixDerived>& A) : Base(A.derived()), m_restart(30),m_neig(0),m_r(0),m_maxNeig(5),m_isDeflAllocated(false),m_isDeflInitialized(false) {}
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138 |
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139 | ~DGMRES() {}
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140 |
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141 | /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A
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142 | * \a x0 as an initial solution.
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143 | *
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144 | * \sa compute()
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145 | */
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146 | template<typename Rhs,typename Guess>
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147 | inline const internal::solve_retval_with_guess<DGMRES, Rhs, Guess>
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148 | solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const
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149 | {
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150 | eigen_assert(m_isInitialized && "DGMRES is not initialized.");
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151 | eigen_assert(Base::rows()==b.rows()
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152 | && "DGMRES::solve(): invalid number of rows of the right hand side matrix b");
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153 | return internal::solve_retval_with_guess
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154 | <DGMRES, Rhs, Guess>(*this, b.derived(), x0);
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155 | }
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156 |
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157 | /** \internal */
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158 | template<typename Rhs,typename Dest>
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159 | void _solveWithGuess(const Rhs& b, Dest& x) const
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160 | {
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161 | bool failed = false;
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162 | for(int j=0; j<b.cols(); ++j)
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163 | {
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164 | m_iterations = Base::maxIterations();
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165 | m_error = Base::m_tolerance;
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166 |
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167 | typename Dest::ColXpr xj(x,j);
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168 | dgmres(*mp_matrix, b.col(j), xj, Base::m_preconditioner);
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169 | }
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170 | m_info = failed ? NumericalIssue
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171 | : m_error <= Base::m_tolerance ? Success
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172 | : NoConvergence;
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173 | m_isInitialized = true;
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174 | }
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175 |
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176 | /** \internal */
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177 | template<typename Rhs,typename Dest>
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178 | void _solve(const Rhs& b, Dest& x) const
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179 | {
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180 | x = b;
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181 | _solveWithGuess(b,x);
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182 | }
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183 | /**
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184 | * Get the restart value
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185 | */
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186 | int restart() { return m_restart; }
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187 |
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188 | /**
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189 | * Set the restart value (default is 30)
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190 | */
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191 | void set_restart(const int restart) { m_restart=restart; }
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192 |
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193 | /**
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194 | * Set the number of eigenvalues to deflate at each restart
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195 | */
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196 | void setEigenv(const int neig)
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197 | {
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198 | m_neig = neig;
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199 | if (neig+1 > m_maxNeig) m_maxNeig = neig+1; // To allow for complex conjugates
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200 | }
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201 |
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202 | /**
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203 | * Get the size of the deflation subspace size
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204 | */
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205 | int deflSize() {return m_r; }
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206 |
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207 | /**
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208 | * Set the maximum size of the deflation subspace
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209 | */
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210 | void setMaxEigenv(const int maxNeig) { m_maxNeig = maxNeig; }
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211 |
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212 | protected:
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213 | // DGMRES algorithm
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214 | template<typename Rhs, typename Dest>
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215 | void dgmres(const MatrixType& mat,const Rhs& rhs, Dest& x, const Preconditioner& precond) const;
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216 | // Perform one cycle of GMRES
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217 | template<typename Dest>
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218 | int dgmresCycle(const MatrixType& mat, const Preconditioner& precond, Dest& x, DenseVector& r0, RealScalar& beta, const RealScalar& normRhs, int& nbIts) const;
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219 | // Compute data to use for deflation
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220 | int dgmresComputeDeflationData(const MatrixType& mat, const Preconditioner& precond, const Index& it, Index& neig) const;
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221 | // Apply deflation to a vector
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222 | template<typename RhsType, typename DestType>
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223 | int dgmresApplyDeflation(const RhsType& In, DestType& Out) const;
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224 | ComplexVector schurValues(const ComplexSchur<DenseMatrix>& schurofH) const;
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225 | ComplexVector schurValues(const RealSchur<DenseMatrix>& schurofH) const;
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226 | // Init data for deflation
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227 | void dgmresInitDeflation(Index& rows) const;
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228 | mutable DenseMatrix m_V; // Krylov basis vectors
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229 | mutable DenseMatrix m_H; // Hessenberg matrix
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230 | mutable DenseMatrix m_Hes; // Initial hessenberg matrix wihout Givens rotations applied
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231 | mutable Index m_restart; // Maximum size of the Krylov subspace
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232 | mutable DenseMatrix m_U; // Vectors that form the basis of the invariant subspace
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233 | mutable DenseMatrix m_MU; // matrix operator applied to m_U (for next cycles)
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234 | mutable DenseMatrix m_T; /* T=U^T*M^{-1}*A*U */
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235 | mutable PartialPivLU<DenseMatrix> m_luT; // LU factorization of m_T
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236 | mutable int m_neig; //Number of eigenvalues to extract at each restart
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237 | mutable int m_r; // Current number of deflated eigenvalues, size of m_U
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238 | mutable int m_maxNeig; // Maximum number of eigenvalues to deflate
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239 | mutable RealScalar m_lambdaN; //Modulus of the largest eigenvalue of A
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240 | mutable bool m_isDeflAllocated;
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241 | mutable bool m_isDeflInitialized;
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242 |
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243 | //Adaptive strategy
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244 | mutable RealScalar m_smv; // Smaller multiple of the remaining number of steps allowed
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245 | mutable bool m_force; // Force the use of deflation at each restart
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246 |
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247 | };
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248 | /**
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249 | * \brief Perform several cycles of restarted GMRES with modified Gram Schmidt,
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250 | *
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251 | * A right preconditioner is used combined with deflation.
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252 | *
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253 | */
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254 | template< typename _MatrixType, typename _Preconditioner>
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255 | template<typename Rhs, typename Dest>
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256 | void DGMRES<_MatrixType, _Preconditioner>::dgmres(const MatrixType& mat,const Rhs& rhs, Dest& x,
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257 | const Preconditioner& precond) const
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258 | {
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259 | //Initialization
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260 | int n = mat.rows();
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261 | DenseVector r0(n);
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262 | int nbIts = 0;
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263 | m_H.resize(m_restart+1, m_restart);
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264 | m_Hes.resize(m_restart, m_restart);
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265 | m_V.resize(n,m_restart+1);
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266 | //Initial residual vector and intial norm
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267 | x = precond.solve(x);
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268 | r0 = rhs - mat * x;
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269 | RealScalar beta = r0.norm();
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270 | RealScalar normRhs = rhs.norm();
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271 | m_error = beta/normRhs;
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272 | if(m_error < m_tolerance)
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273 | m_info = Success;
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274 | else
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275 | m_info = NoConvergence;
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276 |
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277 | // Iterative process
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278 | while (nbIts < m_iterations && m_info == NoConvergence)
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279 | {
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280 | dgmresCycle(mat, precond, x, r0, beta, normRhs, nbIts);
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281 |
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282 | // Compute the new residual vector for the restart
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283 | if (nbIts < m_iterations && m_info == NoConvergence)
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284 | r0 = rhs - mat * x;
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285 | }
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286 | }
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287 |
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288 | /**
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289 | * \brief Perform one restart cycle of DGMRES
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290 | * \param mat The coefficient matrix
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291 | * \param precond The preconditioner
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292 | * \param x the new approximated solution
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293 | * \param r0 The initial residual vector
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294 | * \param beta The norm of the residual computed so far
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295 | * \param normRhs The norm of the right hand side vector
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296 | * \param nbIts The number of iterations
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297 | */
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298 | template< typename _MatrixType, typename _Preconditioner>
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299 | template<typename Dest>
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300 | int DGMRES<_MatrixType, _Preconditioner>::dgmresCycle(const MatrixType& mat, const Preconditioner& precond, Dest& x, DenseVector& r0, RealScalar& beta, const RealScalar& normRhs, int& nbIts) const
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301 | {
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302 | //Initialization
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303 | DenseVector g(m_restart+1); // Right hand side of the least square problem
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304 | g.setZero();
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305 | g(0) = Scalar(beta);
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306 | m_V.col(0) = r0/beta;
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307 | m_info = NoConvergence;
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308 | std::vector<JacobiRotation<Scalar> >gr(m_restart); // Givens rotations
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309 | int it = 0; // Number of inner iterations
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310 | int n = mat.rows();
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311 | DenseVector tv1(n), tv2(n); //Temporary vectors
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312 | while (m_info == NoConvergence && it < m_restart && nbIts < m_iterations)
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313 | {
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314 | // Apply preconditioner(s) at right
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315 | if (m_isDeflInitialized )
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316 | {
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317 | dgmresApplyDeflation(m_V.col(it), tv1); // Deflation
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318 | tv2 = precond.solve(tv1);
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319 | }
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320 | else
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321 | {
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322 | tv2 = precond.solve(m_V.col(it)); // User's selected preconditioner
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323 | }
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324 | tv1 = mat * tv2;
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325 |
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326 | // Orthogonalize it with the previous basis in the basis using modified Gram-Schmidt
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327 | Scalar coef;
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328 | for (int i = 0; i <= it; ++i)
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329 | {
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330 | coef = tv1.dot(m_V.col(i));
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331 | tv1 = tv1 - coef * m_V.col(i);
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332 | m_H(i,it) = coef;
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333 | m_Hes(i,it) = coef;
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334 | }
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335 | // Normalize the vector
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336 | coef = tv1.norm();
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337 | m_V.col(it+1) = tv1/coef;
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338 | m_H(it+1, it) = coef;
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339 | // m_Hes(it+1,it) = coef;
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340 |
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341 | // FIXME Check for happy breakdown
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342 |
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343 | // Update Hessenberg matrix with Givens rotations
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344 | for (int i = 1; i <= it; ++i)
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345 | {
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346 | m_H.col(it).applyOnTheLeft(i-1,i,gr[i-1].adjoint());
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347 | }
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348 | // Compute the new plane rotation
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349 | gr[it].makeGivens(m_H(it, it), m_H(it+1,it));
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350 | // Apply the new rotation
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351 | m_H.col(it).applyOnTheLeft(it,it+1,gr[it].adjoint());
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352 | g.applyOnTheLeft(it,it+1, gr[it].adjoint());
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353 |
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354 | beta = std::abs(g(it+1));
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355 | m_error = beta/normRhs;
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356 | std::cerr << nbIts << " Relative Residual Norm " << m_error << std::endl;
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357 | it++; nbIts++;
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358 |
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359 | if (m_error < m_tolerance)
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360 | {
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361 | // The method has converged
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362 | m_info = Success;
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363 | break;
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364 | }
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365 | }
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366 |
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367 | // Compute the new coefficients by solving the least square problem
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368 | // it++;
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369 | //FIXME Check first if the matrix is singular ... zero diagonal
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370 | DenseVector nrs(m_restart);
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371 | nrs = m_H.topLeftCorner(it,it).template triangularView<Upper>().solve(g.head(it));
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372 |
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373 | // Form the new solution
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374 | if (m_isDeflInitialized)
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375 | {
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376 | tv1 = m_V.leftCols(it) * nrs;
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377 | dgmresApplyDeflation(tv1, tv2);
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378 | x = x + precond.solve(tv2);
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379 | }
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380 | else
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381 | x = x + precond.solve(m_V.leftCols(it) * nrs);
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382 |
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383 | // Go for a new cycle and compute data for deflation
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384 | if(nbIts < m_iterations && m_info == NoConvergence && m_neig > 0 && (m_r+m_neig) < m_maxNeig)
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385 | dgmresComputeDeflationData(mat, precond, it, m_neig);
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386 | return 0;
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387 |
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388 | }
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389 |
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390 |
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391 | template< typename _MatrixType, typename _Preconditioner>
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392 | void DGMRES<_MatrixType, _Preconditioner>::dgmresInitDeflation(Index& rows) const
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393 | {
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394 | m_U.resize(rows, m_maxNeig);
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395 | m_MU.resize(rows, m_maxNeig);
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396 | m_T.resize(m_maxNeig, m_maxNeig);
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397 | m_lambdaN = 0.0;
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398 | m_isDeflAllocated = true;
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399 | }
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400 |
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401 | template< typename _MatrixType, typename _Preconditioner>
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402 | inline typename DGMRES<_MatrixType, _Preconditioner>::ComplexVector DGMRES<_MatrixType, _Preconditioner>::schurValues(const ComplexSchur<DenseMatrix>& schurofH) const
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403 | {
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404 | return schurofH.matrixT().diagonal();
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405 | }
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406 |
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407 | template< typename _MatrixType, typename _Preconditioner>
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408 | inline typename DGMRES<_MatrixType, _Preconditioner>::ComplexVector DGMRES<_MatrixType, _Preconditioner>::schurValues(const RealSchur<DenseMatrix>& schurofH) const
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409 | {
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410 | typedef typename MatrixType::Index Index;
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411 | const DenseMatrix& T = schurofH.matrixT();
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412 | Index it = T.rows();
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413 | ComplexVector eig(it);
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414 | Index j = 0;
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415 | while (j < it-1)
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416 | {
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417 | if (T(j+1,j) ==Scalar(0))
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418 | {
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419 | eig(j) = std::complex<RealScalar>(T(j,j),RealScalar(0));
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420 | j++;
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421 | }
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422 | else
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423 | {
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424 | eig(j) = std::complex<RealScalar>(T(j,j),T(j+1,j));
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425 | eig(j+1) = std::complex<RealScalar>(T(j,j+1),T(j+1,j+1));
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426 | j++;
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427 | }
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428 | }
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429 | if (j < it-1) eig(j) = std::complex<RealScalar>(T(j,j),RealScalar(0));
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430 | return eig;
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431 | }
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432 |
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433 | template< typename _MatrixType, typename _Preconditioner>
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434 | int DGMRES<_MatrixType, _Preconditioner>::dgmresComputeDeflationData(const MatrixType& mat, const Preconditioner& precond, const Index& it, Index& neig) const
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435 | {
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436 | // First, find the Schur form of the Hessenberg matrix H
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437 | typename internal::conditional<NumTraits<Scalar>::IsComplex, ComplexSchur<DenseMatrix>, RealSchur<DenseMatrix> >::type schurofH;
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438 | bool computeU = true;
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439 | DenseMatrix matrixQ(it,it);
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440 | matrixQ.setIdentity();
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441 | schurofH.computeFromHessenberg(m_Hes.topLeftCorner(it,it), matrixQ, computeU);
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442 |
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443 | ComplexVector eig(it);
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444 | Matrix<Index,Dynamic,1>perm(it);
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445 | eig = this->schurValues(schurofH);
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446 |
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447 | // Reorder the absolute values of Schur values
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448 | DenseRealVector modulEig(it);
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449 | for (int j=0; j<it; ++j) modulEig(j) = std::abs(eig(j));
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450 | perm.setLinSpaced(it,0,it-1);
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451 | internal::sortWithPermutation(modulEig, perm, neig);
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452 |
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453 | if (!m_lambdaN)
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454 | {
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455 | m_lambdaN = (std::max)(modulEig.maxCoeff(), m_lambdaN);
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456 | }
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457 | //Count the real number of extracted eigenvalues (with complex conjugates)
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458 | int nbrEig = 0;
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459 | while (nbrEig < neig)
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460 | {
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461 | if(eig(perm(it-nbrEig-1)).imag() == RealScalar(0)) nbrEig++;
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462 | else nbrEig += 2;
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463 | }
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464 | // Extract the Schur vectors corresponding to the smallest Ritz values
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465 | DenseMatrix Sr(it, nbrEig);
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466 | Sr.setZero();
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467 | for (int j = 0; j < nbrEig; j++)
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468 | {
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469 | Sr.col(j) = schurofH.matrixU().col(perm(it-j-1));
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470 | }
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471 |
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472 | // Form the Schur vectors of the initial matrix using the Krylov basis
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473 | DenseMatrix X;
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474 | X = m_V.leftCols(it) * Sr;
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475 | if (m_r)
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476 | {
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477 | // Orthogonalize X against m_U using modified Gram-Schmidt
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478 | for (int j = 0; j < nbrEig; j++)
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479 | for (int k =0; k < m_r; k++)
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480 | X.col(j) = X.col(j) - (m_U.col(k).dot(X.col(j)))*m_U.col(k);
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481 | }
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482 |
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483 | // Compute m_MX = A * M^-1 * X
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484 | Index m = m_V.rows();
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485 | if (!m_isDeflAllocated)
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486 | dgmresInitDeflation(m);
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487 | DenseMatrix MX(m, nbrEig);
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488 | DenseVector tv1(m);
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489 | for (int j = 0; j < nbrEig; j++)
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490 | {
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491 | tv1 = mat * X.col(j);
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492 | MX.col(j) = precond.solve(tv1);
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493 | }
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494 |
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495 | //Update m_T = [U'MU U'MX; X'MU X'MX]
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496 | m_T.block(m_r, m_r, nbrEig, nbrEig) = X.transpose() * MX;
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497 | if(m_r)
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498 | {
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499 | m_T.block(0, m_r, m_r, nbrEig) = m_U.leftCols(m_r).transpose() * MX;
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500 | m_T.block(m_r, 0, nbrEig, m_r) = X.transpose() * m_MU.leftCols(m_r);
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501 | }
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502 |
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503 | // Save X into m_U and m_MX in m_MU
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504 | for (int j = 0; j < nbrEig; j++) m_U.col(m_r+j) = X.col(j);
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505 | for (int j = 0; j < nbrEig; j++) m_MU.col(m_r+j) = MX.col(j);
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506 | // Increase the size of the invariant subspace
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507 | m_r += nbrEig;
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508 |
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509 | // Factorize m_T into m_luT
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510 | m_luT.compute(m_T.topLeftCorner(m_r, m_r));
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511 |
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512 | //FIXME CHeck if the factorization was correctly done (nonsingular matrix)
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513 | m_isDeflInitialized = true;
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514 | return 0;
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515 | }
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516 | template<typename _MatrixType, typename _Preconditioner>
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517 | template<typename RhsType, typename DestType>
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518 | int DGMRES<_MatrixType, _Preconditioner>::dgmresApplyDeflation(const RhsType &x, DestType &y) const
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519 | {
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520 | DenseVector x1 = m_U.leftCols(m_r).transpose() * x;
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521 | y = x + m_U.leftCols(m_r) * ( m_lambdaN * m_luT.solve(x1) - x1);
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522 | return 0;
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523 | }
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524 |
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525 | namespace internal {
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526 |
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527 | template<typename _MatrixType, typename _Preconditioner, typename Rhs>
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528 | struct solve_retval<DGMRES<_MatrixType, _Preconditioner>, Rhs>
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529 | : solve_retval_base<DGMRES<_MatrixType, _Preconditioner>, Rhs>
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530 | {
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531 | typedef DGMRES<_MatrixType, _Preconditioner> Dec;
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532 | EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs)
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533 |
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534 | template<typename Dest> void evalTo(Dest& dst) const
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535 | {
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536 | dec()._solve(rhs(),dst);
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537 | }
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538 | };
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539 | } // end namespace internal
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540 |
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541 | } // end namespace Eigen
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542 | #endif
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