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289 lines
13 KiB
289 lines
13 KiB
// This file is part of Eigen, a lightweight C++ template library
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// for linear algebra.
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//
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// Copyright (C) 2012 Giacomo Po <gpo@ucla.edu>
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// Copyright (C) 2011-2014 Gael Guennebaud <gael.guennebaud@inria.fr>
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//
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// This Source Code Form is subject to the terms of the Mozilla
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// Public License v. 2.0. If a copy of the MPL was not distributed
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// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
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#ifndef EIGEN_MINRES_H_
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#define EIGEN_MINRES_H_
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namespace Eigen {
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namespace internal {
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/** \internal Low-level MINRES algorithm
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* \param mat The matrix A
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* \param rhs The right hand side vector b
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* \param x On input and initial solution, on output the computed solution.
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* \param precond A right preconditioner being able to efficiently solve for an
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* approximation of Ax=b (regardless of b)
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* \param iters On input the max number of iteration, on output the number of performed iterations.
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* \param tol_error On input the tolerance error, on output an estimation of the relative error.
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*/
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template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
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EIGEN_DONT_INLINE
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void minres(const MatrixType& mat, const Rhs& rhs, Dest& x,
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const Preconditioner& precond, Index& iters,
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typename Dest::RealScalar& tol_error)
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{
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using std::sqrt;
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typedef typename Dest::RealScalar RealScalar;
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typedef typename Dest::Scalar Scalar;
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typedef Matrix<Scalar,Dynamic,1> VectorType;
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// Check for zero rhs
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const RealScalar rhsNorm2(rhs.squaredNorm());
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if(rhsNorm2 == 0)
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{
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x.setZero();
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iters = 0;
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tol_error = 0;
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return;
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}
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// initialize
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const Index maxIters(iters); // initialize maxIters to iters
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const Index N(mat.cols()); // the size of the matrix
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const RealScalar threshold2(tol_error*tol_error*rhsNorm2); // convergence threshold (compared to residualNorm2)
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// Initialize preconditioned Lanczos
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VectorType v_old(N); // will be initialized inside loop
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VectorType v( VectorType::Zero(N) ); //initialize v
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VectorType v_new(rhs-mat*x); //initialize v_new
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RealScalar residualNorm2(v_new.squaredNorm());
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VectorType w(N); // will be initialized inside loop
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VectorType w_new(precond.solve(v_new)); // initialize w_new
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// RealScalar beta; // will be initialized inside loop
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RealScalar beta_new2(v_new.dot(w_new));
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eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE");
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RealScalar beta_new(sqrt(beta_new2));
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const RealScalar beta_one(beta_new);
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v_new /= beta_new;
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w_new /= beta_new;
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// Initialize other variables
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RealScalar c(1.0); // the cosine of the Givens rotation
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RealScalar c_old(1.0);
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RealScalar s(0.0); // the sine of the Givens rotation
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RealScalar s_old(0.0); // the sine of the Givens rotation
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VectorType p_oold(N); // will be initialized in loop
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VectorType p_old(VectorType::Zero(N)); // initialize p_old=0
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VectorType p(p_old); // initialize p=0
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RealScalar eta(1.0);
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iters = 0; // reset iters
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while ( iters < maxIters )
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{
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// Preconditioned Lanczos
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/* Note that there are 4 variants on the Lanczos algorithm. These are
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* described in Paige, C. C. (1972). Computational variants of
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* the Lanczos method for the eigenproblem. IMA Journal of Applied
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* Mathematics, 10(3), 373–381. The current implementation corresponds
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* to the case A(2,7) in the paper. It also corresponds to
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* algorithm 6.14 in Y. Saad, Iterative Methods for Sparse Linear
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* Systems, 2003 p.173. For the preconditioned version see
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* A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM (1987).
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*/
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const RealScalar beta(beta_new);
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v_old = v; // update: at first time step, this makes v_old = 0 so value of beta doesn't matter
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// const VectorType v_old(v); // NOT SURE IF CREATING v_old EVERY ITERATION IS EFFICIENT
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v = v_new; // update
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w = w_new; // update
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// const VectorType w(w_new); // NOT SURE IF CREATING w EVERY ITERATION IS EFFICIENT
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v_new.noalias() = mat*w - beta*v_old; // compute v_new
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const RealScalar alpha = v_new.dot(w);
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v_new -= alpha*v; // overwrite v_new
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w_new = precond.solve(v_new); // overwrite w_new
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beta_new2 = v_new.dot(w_new); // compute beta_new
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eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE");
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beta_new = sqrt(beta_new2); // compute beta_new
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v_new /= beta_new; // overwrite v_new for next iteration
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w_new /= beta_new; // overwrite w_new for next iteration
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// Givens rotation
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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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const RealScalar r3 =s_old*beta; // s, s_old, c and c_old are still from previous iteration
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const RealScalar r1_hat=c*alpha-c_old*s*beta;
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const RealScalar r1 =sqrt( std::pow(r1_hat,2) + std::pow(beta_new,2) );
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c_old = c; // store for next iteration
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s_old = s; // store for next iteration
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c=r1_hat/r1; // new cosine
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s=beta_new/r1; // new sine
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// Update solution
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p_oold = p_old;
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// const VectorType p_oold(p_old); // NOT SURE IF CREATING p_oold EVERY ITERATION IS EFFICIENT
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p_old = p;
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p.noalias()=(w-r2*p_old-r3*p_oold) /r1; // IS NOALIAS REQUIRED?
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x += beta_one*c*eta*p;
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/* Update the squared residual. Note that this is the estimated residual.
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The real residual |Ax-b|^2 may be slightly larger */
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residualNorm2 *= s*s;
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if ( residualNorm2 < threshold2)
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{
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break;
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}
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eta=-s*eta; // update eta
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iters++; // increment iteration number (for output purposes)
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}
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/* Compute error. Note that this is the estimated error. The real
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error |Ax-b|/|b| may be slightly larger */
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tol_error = std::sqrt(residualNorm2 / rhsNorm2);
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}
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}
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template< typename _MatrixType, int _UpLo=Lower,
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typename _Preconditioner = IdentityPreconditioner>
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class MINRES;
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namespace internal {
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template< typename _MatrixType, int _UpLo, typename _Preconditioner>
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struct traits<MINRES<_MatrixType,_UpLo,_Preconditioner> >
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{
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typedef _MatrixType MatrixType;
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typedef _Preconditioner Preconditioner;
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};
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}
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/** \ingroup IterativeLinearSolvers_Module
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* \brief A minimal residual solver for sparse symmetric problems
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*
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* This class allows to solve for A.x = b sparse linear problems using the MINRES algorithm
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* of Paige and Saunders (1975). The sparse matrix A must be symmetric (possibly indefinite).
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* The vectors x and b can be either dense or sparse.
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*
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* \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix.
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* \tparam _UpLo the triangular part that will be used for the computations. It can be Lower,
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* Upper, or Lower|Upper in which the full matrix entries will be considered. Default is Lower.
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* \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner
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*
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* The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
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* and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations
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* and NumTraits<Scalar>::epsilon() for the tolerance.
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*
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* This class can be used as the direct solver classes. Here is a typical usage example:
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* \code
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* int n = 10000;
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* VectorXd x(n), b(n);
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* SparseMatrix<double> A(n,n);
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* // fill A and b
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* MINRES<SparseMatrix<double> > mr;
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* mr.compute(A);
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* x = mr.solve(b);
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* std::cout << "#iterations: " << mr.iterations() << std::endl;
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* std::cout << "estimated error: " << mr.error() << std::endl;
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* // update b, and solve again
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* x = mr.solve(b);
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* \endcode
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*
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* By default the iterations start with x=0 as an initial guess of the solution.
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* One can control the start using the solveWithGuess() method.
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*
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* MINRES can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink.
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*
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* \sa class ConjugateGradient, BiCGSTAB, SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
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*/
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template< typename _MatrixType, int _UpLo, typename _Preconditioner>
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class MINRES : public IterativeSolverBase<MINRES<_MatrixType,_UpLo,_Preconditioner> >
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{
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typedef IterativeSolverBase<MINRES> Base;
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using Base::matrix;
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using Base::m_error;
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using Base::m_iterations;
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using Base::m_info;
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using Base::m_isInitialized;
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public:
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using Base::_solve_impl;
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typedef _MatrixType MatrixType;
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typedef typename MatrixType::Scalar Scalar;
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typedef typename MatrixType::RealScalar RealScalar;
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typedef _Preconditioner Preconditioner;
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enum {UpLo = _UpLo};
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public:
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/** Default constructor. */
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MINRES() : Base() {}
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/** Initialize the solver with matrix \a A for further \c Ax=b solving.
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*
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* This constructor is a shortcut for the default constructor followed
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* by a call to compute().
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*
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* \warning this class stores a reference to the matrix A as well as some
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* precomputed values that depend on it. Therefore, if \a A is changed
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* this class becomes invalid. Call compute() to update it with the new
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* matrix A, or modify a copy of A.
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*/
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template<typename MatrixDerived>
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explicit MINRES(const EigenBase<MatrixDerived>& A) : Base(A.derived()) {}
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/** Destructor. */
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~MINRES(){}
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/** \internal */
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template<typename Rhs,typename Dest>
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void _solve_with_guess_impl(const Rhs& b, Dest& x) const
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{
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typedef typename Base::MatrixWrapper MatrixWrapper;
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typedef typename Base::ActualMatrixType ActualMatrixType;
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enum {
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TransposeInput = (!MatrixWrapper::MatrixFree)
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&& (UpLo==(Lower|Upper))
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&& (!MatrixType::IsRowMajor)
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&& (!NumTraits<Scalar>::IsComplex)
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};
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typedef typename internal::conditional<TransposeInput,Transpose<const ActualMatrixType>, ActualMatrixType const&>::type RowMajorWrapper;
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EIGEN_STATIC_ASSERT(EIGEN_IMPLIES(MatrixWrapper::MatrixFree,UpLo==(Lower|Upper)),MATRIX_FREE_CONJUGATE_GRADIENT_IS_COMPATIBLE_WITH_UPPER_UNION_LOWER_MODE_ONLY);
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typedef typename internal::conditional<UpLo==(Lower|Upper),
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RowMajorWrapper,
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typename MatrixWrapper::template ConstSelfAdjointViewReturnType<UpLo>::Type
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>::type SelfAdjointWrapper;
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m_iterations = Base::maxIterations();
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m_error = Base::m_tolerance;
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RowMajorWrapper row_mat(matrix());
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for(int j=0; j<b.cols(); ++j)
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{
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m_iterations = Base::maxIterations();
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m_error = Base::m_tolerance;
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typename Dest::ColXpr xj(x,j);
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internal::minres(SelfAdjointWrapper(row_mat), b.col(j), xj,
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Base::m_preconditioner, m_iterations, m_error);
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}
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m_isInitialized = true;
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m_info = m_error <= Base::m_tolerance ? Success : NoConvergence;
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}
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/** \internal */
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template<typename Rhs,typename Dest>
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void _solve_impl(const Rhs& b, MatrixBase<Dest> &x) const
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{
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x.setZero();
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_solve_with_guess_impl(b,x.derived());
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}
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protected:
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};
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} // end namespace Eigen
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#endif // EIGEN_MINRES_H
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