[136] | 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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