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 Giacomo Po <gpo@ucla.edu>
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5 | // Copyright (C) 2011 Gael Guennebaud <gael.guennebaud@inria.fr>
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6 | //
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7 | // This Source Code Form is subject to the terms of the Mozilla
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8 | // Public License v. 2.0. If a copy of the MPL was not distributed
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9 | // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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10 |
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11 |
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12 | #ifndef EIGEN_MINRES_H_
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13 | #define EIGEN_MINRES_H_
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14 |
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15 |
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16 | namespace Eigen {
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17 |
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18 | namespace internal {
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19 |
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20 | /** \internal Low-level MINRES algorithm
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21 | * \param mat The matrix A
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22 | * \param rhs The right hand side vector b
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23 | * \param x On input and initial solution, on output the computed solution.
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24 | * \param precond A right preconditioner being able to efficiently solve for an
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25 | * approximation of Ax=b (regardless of b)
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26 | * \param iters On input the max number of iteration, on output the number of performed iterations.
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27 | * \param tol_error On input the tolerance error, on output an estimation of the relative error.
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28 | */
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29 | template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
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30 | EIGEN_DONT_INLINE
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31 | void minres(const MatrixType& mat, const Rhs& rhs, Dest& x,
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32 | const Preconditioner& precond, int& iters,
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33 | typename Dest::RealScalar& tol_error)
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34 | {
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35 | using std::sqrt;
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36 | typedef typename Dest::RealScalar RealScalar;
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37 | typedef typename Dest::Scalar Scalar;
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38 | typedef Matrix<Scalar,Dynamic,1> VectorType;
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39 |
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40 | // Check for zero rhs
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41 | const RealScalar rhsNorm2(rhs.squaredNorm());
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42 | if(rhsNorm2 == 0)
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43 | {
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44 | x.setZero();
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45 | iters = 0;
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46 | tol_error = 0;
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47 | return;
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48 | }
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49 |
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50 | // initialize
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51 | const int maxIters(iters); // initialize maxIters to iters
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52 | const int N(mat.cols()); // the size of the matrix
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53 | const RealScalar threshold2(tol_error*tol_error*rhsNorm2); // convergence threshold (compared to residualNorm2)
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54 |
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55 | // Initialize preconditioned Lanczos
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56 | VectorType v_old(N); // will be initialized inside loop
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57 | VectorType v( VectorType::Zero(N) ); //initialize v
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58 | VectorType v_new(rhs-mat*x); //initialize v_new
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59 | RealScalar residualNorm2(v_new.squaredNorm());
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60 | VectorType w(N); // will be initialized inside loop
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61 | VectorType w_new(precond.solve(v_new)); // initialize w_new
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62 | // RealScalar beta; // will be initialized inside loop
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63 | RealScalar beta_new2(v_new.dot(w_new));
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64 | eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE");
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65 | RealScalar beta_new(sqrt(beta_new2));
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66 | const RealScalar beta_one(beta_new);
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67 | v_new /= beta_new;
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68 | w_new /= beta_new;
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69 | // Initialize other variables
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70 | RealScalar c(1.0); // the cosine of the Givens rotation
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71 | RealScalar c_old(1.0);
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72 | RealScalar s(0.0); // the sine of the Givens rotation
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73 | RealScalar s_old(0.0); // the sine of the Givens rotation
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74 | VectorType p_oold(N); // will be initialized in loop
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75 | VectorType p_old(VectorType::Zero(N)); // initialize p_old=0
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76 | VectorType p(p_old); // initialize p=0
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77 | RealScalar eta(1.0);
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78 |
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79 | iters = 0; // reset iters
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80 | while ( iters < maxIters )
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81 | {
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82 | // Preconditioned Lanczos
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83 | /* Note that there are 4 variants on the Lanczos algorithm. These are
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84 | * described in Paige, C. C. (1972). Computational variants of
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85 | * the Lanczos method for the eigenproblem. IMA Journal of Applied
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86 | * Mathematics, 10(3), 373–381. The current implementation corresponds
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87 | * to the case A(2,7) in the paper. It also corresponds to
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88 | * algorithm 6.14 in Y. Saad, Iterative Methods for Sparse Linear
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89 | * Systems, 2003 p.173. For the preconditioned version see
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90 | * A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM (1987).
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91 | */
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92 | const RealScalar beta(beta_new);
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93 | v_old = v; // update: at first time step, this makes v_old = 0 so value of beta doesn't matter
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94 | // const VectorType v_old(v); // NOT SURE IF CREATING v_old EVERY ITERATION IS EFFICIENT
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95 | v = v_new; // update
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96 | w = w_new; // update
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97 | // const VectorType w(w_new); // NOT SURE IF CREATING w EVERY ITERATION IS EFFICIENT
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98 | v_new.noalias() = mat*w - beta*v_old; // compute v_new
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99 | const RealScalar alpha = v_new.dot(w);
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100 | v_new -= alpha*v; // overwrite v_new
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101 | w_new = precond.solve(v_new); // overwrite w_new
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102 | beta_new2 = v_new.dot(w_new); // compute beta_new
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103 | eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE");
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104 | beta_new = sqrt(beta_new2); // compute beta_new
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105 | v_new /= beta_new; // overwrite v_new for next iteration
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106 | w_new /= beta_new; // overwrite w_new for next iteration
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107 |
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108 | // Givens rotation
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109 | const RealScalar r2 =s*alpha+c*c_old*beta; // s, s_old, c and c_old are still from previous iteration
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110 | const RealScalar r3 =s_old*beta; // s, s_old, c and c_old are still from previous iteration
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111 | const RealScalar r1_hat=c*alpha-c_old*s*beta;
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112 | const RealScalar r1 =sqrt( std::pow(r1_hat,2) + std::pow(beta_new,2) );
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113 | c_old = c; // store for next iteration
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114 | s_old = s; // store for next iteration
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115 | c=r1_hat/r1; // new cosine
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116 | s=beta_new/r1; // new sine
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117 |
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118 | // Update solution
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119 | p_oold = p_old;
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120 | // const VectorType p_oold(p_old); // NOT SURE IF CREATING p_oold EVERY ITERATION IS EFFICIENT
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121 | p_old = p;
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122 | p.noalias()=(w-r2*p_old-r3*p_oold) /r1; // IS NOALIAS REQUIRED?
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123 | x += beta_one*c*eta*p;
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124 |
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125 | /* Update the squared residual. Note that this is the estimated residual.
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126 | The real residual |Ax-b|^2 may be slightly larger */
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127 | residualNorm2 *= s*s;
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128 |
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129 | if ( residualNorm2 < threshold2)
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130 | {
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131 | break;
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132 | }
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133 |
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134 | eta=-s*eta; // update eta
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135 | iters++; // increment iteration number (for output purposes)
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136 | }
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137 |
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138 | /* Compute error. Note that this is the estimated error. The real
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139 | error |Ax-b|/|b| may be slightly larger */
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140 | tol_error = std::sqrt(residualNorm2 / rhsNorm2);
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141 | }
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142 |
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143 | }
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144 |
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145 | template< typename _MatrixType, int _UpLo=Lower,
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146 | typename _Preconditioner = IdentityPreconditioner>
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147 | // typename _Preconditioner = IdentityPreconditioner<typename _MatrixType::Scalar> > // preconditioner must be positive definite
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148 | class MINRES;
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149 |
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150 | namespace internal {
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151 |
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152 | template< typename _MatrixType, int _UpLo, typename _Preconditioner>
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153 | struct traits<MINRES<_MatrixType,_UpLo,_Preconditioner> >
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154 | {
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155 | typedef _MatrixType MatrixType;
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156 | typedef _Preconditioner Preconditioner;
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157 | };
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158 |
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159 | }
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160 |
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161 | /** \ingroup IterativeLinearSolvers_Module
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162 | * \brief A minimal residual solver for sparse symmetric problems
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163 | *
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164 | * This class allows to solve for A.x = b sparse linear problems using the MINRES algorithm
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165 | * of Paige and Saunders (1975). The sparse matrix A must be symmetric (possibly indefinite).
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166 | * The vectors x and b can be either dense or sparse.
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167 | *
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168 | * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix.
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169 | * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower
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170 | * or Upper. Default is Lower.
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171 | * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
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172 | *
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173 | * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
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174 | * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations
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175 | * and NumTraits<Scalar>::epsilon() for the tolerance.
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176 | *
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177 | * This class can be used as the direct solver classes. Here is a typical usage example:
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178 | * \code
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179 | * int n = 10000;
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180 | * VectorXd x(n), b(n);
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181 | * SparseMatrix<double> A(n,n);
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182 | * // fill A and b
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183 | * MINRES<SparseMatrix<double> > mr;
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184 | * mr.compute(A);
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185 | * x = mr.solve(b);
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186 | * std::cout << "#iterations: " << mr.iterations() << std::endl;
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187 | * std::cout << "estimated error: " << mr.error() << std::endl;
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188 | * // update b, and solve again
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189 | * x = mr.solve(b);
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190 | * \endcode
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191 | *
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192 | * By default the iterations start with x=0 as an initial guess of the solution.
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193 | * One can control the start using the solveWithGuess() method.
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194 | *
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195 | * \sa class ConjugateGradient, BiCGSTAB, SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
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196 | */
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197 | template< typename _MatrixType, int _UpLo, typename _Preconditioner>
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198 | class MINRES : public IterativeSolverBase<MINRES<_MatrixType,_UpLo,_Preconditioner> >
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199 | {
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200 |
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201 | typedef IterativeSolverBase<MINRES> Base;
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202 | using Base::mp_matrix;
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203 | using Base::m_error;
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204 | using Base::m_iterations;
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205 | using Base::m_info;
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206 | using Base::m_isInitialized;
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207 | public:
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208 | typedef _MatrixType MatrixType;
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209 | typedef typename MatrixType::Scalar Scalar;
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210 | typedef typename MatrixType::Index Index;
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211 | typedef typename MatrixType::RealScalar RealScalar;
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212 | typedef _Preconditioner Preconditioner;
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213 |
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214 | enum {UpLo = _UpLo};
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215 |
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216 | public:
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217 |
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218 | /** Default constructor. */
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219 | MINRES() : Base() {}
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220 |
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221 | /** Initialize the solver with matrix \a A for further \c Ax=b solving.
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222 | *
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223 | * This constructor is a shortcut for the default constructor followed
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224 | * by a call to compute().
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225 | *
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226 | * \warning this class stores a reference to the matrix A as well as some
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227 | * precomputed values that depend on it. Therefore, if \a A is changed
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228 | * this class becomes invalid. Call compute() to update it with the new
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229 | * matrix A, or modify a copy of A.
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230 | */
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231 | template<typename MatrixDerived>
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232 | explicit MINRES(const EigenBase<MatrixDerived>& A) : Base(A.derived()) {}
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233 |
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234 | /** Destructor. */
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235 | ~MINRES(){}
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236 |
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237 | /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A
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238 | * \a x0 as an initial solution.
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239 | *
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240 | * \sa compute()
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241 | */
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242 | template<typename Rhs,typename Guess>
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243 | inline const internal::solve_retval_with_guess<MINRES, Rhs, Guess>
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244 | solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const
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245 | {
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246 | eigen_assert(m_isInitialized && "MINRES is not initialized.");
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247 | eigen_assert(Base::rows()==b.rows()
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248 | && "MINRES::solve(): invalid number of rows of the right hand side matrix b");
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249 | return internal::solve_retval_with_guess
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250 | <MINRES, Rhs, Guess>(*this, b.derived(), x0);
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251 | }
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252 |
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253 | /** \internal */
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254 | template<typename Rhs,typename Dest>
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255 | void _solveWithGuess(const Rhs& b, Dest& x) const
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256 | {
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257 | typedef typename internal::conditional<UpLo==(Lower|Upper),
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258 | const MatrixType&,
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259 | SparseSelfAdjointView<const MatrixType, UpLo>
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260 | >::type MatrixWrapperType;
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261 |
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262 | m_iterations = Base::maxIterations();
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263 | m_error = Base::m_tolerance;
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264 |
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265 | for(int j=0; j<b.cols(); ++j)
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266 | {
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267 | m_iterations = Base::maxIterations();
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268 | m_error = Base::m_tolerance;
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269 |
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270 | typename Dest::ColXpr xj(x,j);
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271 | internal::minres(MatrixWrapperType(*mp_matrix), b.col(j), xj,
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272 | Base::m_preconditioner, m_iterations, m_error);
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273 | }
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274 |
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275 | m_isInitialized = true;
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276 | m_info = m_error <= Base::m_tolerance ? Success : NoConvergence;
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277 | }
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278 |
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279 | /** \internal */
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280 | template<typename Rhs,typename Dest>
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281 | void _solve(const Rhs& b, Dest& x) const
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282 | {
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283 | x.setZero();
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284 | _solveWithGuess(b,x);
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285 | }
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286 |
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287 | protected:
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288 |
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289 | };
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290 |
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291 | namespace internal {
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292 |
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293 | template<typename _MatrixType, int _UpLo, typename _Preconditioner, typename Rhs>
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294 | struct solve_retval<MINRES<_MatrixType,_UpLo,_Preconditioner>, Rhs>
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295 | : solve_retval_base<MINRES<_MatrixType,_UpLo,_Preconditioner>, Rhs>
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296 | {
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297 | typedef MINRES<_MatrixType,_UpLo,_Preconditioner> Dec;
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298 | EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs)
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299 |
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300 | template<typename Dest> void evalTo(Dest& dst) const
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301 | {
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302 | dec()._solve(rhs(),dst);
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303 | }
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304 | };
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305 |
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306 | } // end namespace internal
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307 |
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308 | } // end namespace Eigen
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309 |
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310 | #endif // EIGEN_MINRES_H
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311 |
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