| // This file is part of Eigen, a lightweight C++ template library | |
| // for linear algebra. | |
| // | |
| // Copyright (C) 2012 Giacomo Po <gpo@ucla.edu> | |
| // Copyright (C) 2011-2014 Gael Guennebaud <gael.guennebaud@inria.fr> | |
| // Copyright (C) 2018 David Hyde <dabh@stanford.edu> | |
| // | |
| // This Source Code Form is subject to the terms of the Mozilla | |
| // Public License v. 2.0. If a copy of the MPL was not distributed | |
| // with this file, You can obtain one at http://mozilla.org/MPL/2.0/. | |
| namespace Eigen { | |
| namespace internal { | |
| /** \internal Low-level MINRES algorithm | |
| * \param mat The matrix A | |
| * \param rhs The right hand side vector b | |
| * \param x On input and initial solution, on output the computed solution. | |
| * \param precond A right preconditioner being able to efficiently solve for an | |
| * approximation of Ax=b (regardless of b) | |
| * \param iters On input the max number of iteration, on output the number of performed iterations. | |
| * \param tol_error On input the tolerance error, on output an estimation of the relative error. | |
| */ | |
| template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner> | |
| EIGEN_DONT_INLINE | |
| void minres(const MatrixType& mat, const Rhs& rhs, Dest& x, | |
| const Preconditioner& precond, Index& iters, | |
| typename Dest::RealScalar& tol_error) | |
| { | |
| using std::sqrt; | |
| typedef typename Dest::RealScalar RealScalar; | |
| typedef typename Dest::Scalar Scalar; | |
| typedef Matrix<Scalar,Dynamic,1> VectorType; | |
| // Check for zero rhs | |
| const RealScalar rhsNorm2(rhs.squaredNorm()); | |
| if(rhsNorm2 == 0) | |
| { | |
| x.setZero(); | |
| iters = 0; | |
| tol_error = 0; | |
| return; | |
| } | |
| // initialize | |
| const Index maxIters(iters); // initialize maxIters to iters | |
| const Index N(mat.cols()); // the size of the matrix | |
| const RealScalar threshold2(tol_error*tol_error*rhsNorm2); // convergence threshold (compared to residualNorm2) | |
| // Initialize preconditioned Lanczos | |
| VectorType v_old(N); // will be initialized inside loop | |
| VectorType v( VectorType::Zero(N) ); //initialize v | |
| VectorType v_new(rhs-mat*x); //initialize v_new | |
| RealScalar residualNorm2(v_new.squaredNorm()); | |
| VectorType w(N); // will be initialized inside loop | |
| VectorType w_new(precond.solve(v_new)); // initialize w_new | |
| // RealScalar beta; // will be initialized inside loop | |
| RealScalar beta_new2(v_new.dot(w_new)); | |
| eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE"); | |
| RealScalar beta_new(sqrt(beta_new2)); | |
| const RealScalar beta_one(beta_new); | |
| // Initialize other variables | |
| RealScalar c(1.0); // the cosine of the Givens rotation | |
| RealScalar c_old(1.0); | |
| RealScalar s(0.0); // the sine of the Givens rotation | |
| RealScalar s_old(0.0); // the sine of the Givens rotation | |
| VectorType p_oold(N); // will be initialized in loop | |
| VectorType p_old(VectorType::Zero(N)); // initialize p_old=0 | |
| VectorType p(p_old); // initialize p=0 | |
| RealScalar eta(1.0); | |
| iters = 0; // reset iters | |
| while ( iters < maxIters ) | |
| { | |
| // Preconditioned Lanczos | |
| /* Note that there are 4 variants on the Lanczos algorithm. These are | |
| * described in Paige, C. C. (1972). Computational variants of | |
| * the Lanczos method for the eigenproblem. IMA Journal of Applied | |
| * Mathematics, 10(3), 373-381. The current implementation corresponds | |
| * to the case A(2,7) in the paper. It also corresponds to | |
| * algorithm 6.14 in Y. Saad, Iterative Methods for Sparse Linear | |
| * Systems, 2003 p.173. For the preconditioned version see | |
| * A. Greenbaum, Iterative Methods for Solving Linear Systems, SIAM (1987). | |
| */ | |
| const RealScalar beta(beta_new); | |
| v_old = v; // update: at first time step, this makes v_old = 0 so value of beta doesn't matter | |
| v_new /= beta_new; // overwrite v_new for next iteration | |
| w_new /= beta_new; // overwrite w_new for next iteration | |
| v = v_new; // update | |
| w = w_new; // update | |
| v_new.noalias() = mat*w - beta*v_old; // compute v_new | |
| const RealScalar alpha = v_new.dot(w); | |
| v_new -= alpha*v; // overwrite v_new | |
| w_new = precond.solve(v_new); // overwrite w_new | |
| beta_new2 = v_new.dot(w_new); // compute beta_new | |
| eigen_assert(beta_new2 >= 0.0 && "PRECONDITIONER IS NOT POSITIVE DEFINITE"); | |
| beta_new = sqrt(beta_new2); // compute beta_new | |
| // Givens rotation | |
| const RealScalar r2 =s*alpha+c*c_old*beta; // s, s_old, c and c_old are still from previous iteration | |
| const RealScalar r3 =s_old*beta; // s, s_old, c and c_old are still from previous iteration | |
| const RealScalar r1_hat=c*alpha-c_old*s*beta; | |
| const RealScalar r1 =sqrt( std::pow(r1_hat,2) + std::pow(beta_new,2) ); | |
| c_old = c; // store for next iteration | |
| s_old = s; // store for next iteration | |
| c=r1_hat/r1; // new cosine | |
| s=beta_new/r1; // new sine | |
| // Update solution | |
| p_oold = p_old; | |
| p_old = p; | |
| p.noalias()=(w-r2*p_old-r3*p_oold) /r1; // IS NOALIAS REQUIRED? | |
| x += beta_one*c*eta*p; | |
| /* Update the squared residual. Note that this is the estimated residual. | |
| The real residual |Ax-b|^2 may be slightly larger */ | |
| residualNorm2 *= s*s; | |
| if ( residualNorm2 < threshold2) | |
| { | |
| break; | |
| } | |
| eta=-s*eta; // update eta | |
| iters++; // increment iteration number (for output purposes) | |
| } | |
| /* Compute error. Note that this is the estimated error. The real | |
| error |Ax-b|/|b| may be slightly larger */ | |
| tol_error = std::sqrt(residualNorm2 / rhsNorm2); | |
| } | |
| } | |
| template< typename _MatrixType, int _UpLo=Lower, | |
| typename _Preconditioner = IdentityPreconditioner> | |
| class MINRES; | |
| namespace internal { | |
| template< typename _MatrixType, int _UpLo, typename _Preconditioner> | |
| struct traits<MINRES<_MatrixType,_UpLo,_Preconditioner> > | |
| { | |
| typedef _MatrixType MatrixType; | |
| typedef _Preconditioner Preconditioner; | |
| }; | |
| } | |
| /** \ingroup IterativeLinearSolvers_Module | |
| * \brief A minimal residual solver for sparse symmetric problems | |
| * | |
| * This class allows to solve for A.x = b sparse linear problems using the MINRES algorithm | |
| * of Paige and Saunders (1975). The sparse matrix A must be symmetric (possibly indefinite). | |
| * The vectors x and b can be either dense or sparse. | |
| * | |
| * \tparam _MatrixType the type of the sparse matrix A, can be a dense or a sparse matrix. | |
| * \tparam _UpLo the triangular part that will be used for the computations. It can be Lower, | |
| * Upper, or Lower|Upper in which the full matrix entries will be considered. Default is Lower. | |
| * \tparam _Preconditioner the type of the preconditioner. Default is DiagonalPreconditioner | |
| * | |
| * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations() | |
| * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations | |
| * and NumTraits<Scalar>::epsilon() for the tolerance. | |
| * | |
| * This class can be used as the direct solver classes. Here is a typical usage example: | |
| * \code | |
| * int n = 10000; | |
| * VectorXd x(n), b(n); | |
| * SparseMatrix<double> A(n,n); | |
| * // fill A and b | |
| * MINRES<SparseMatrix<double> > mr; | |
| * mr.compute(A); | |
| * x = mr.solve(b); | |
| * std::cout << "#iterations: " << mr.iterations() << std::endl; | |
| * std::cout << "estimated error: " << mr.error() << std::endl; | |
| * // update b, and solve again | |
| * x = mr.solve(b); | |
| * \endcode | |
| * | |
| * By default the iterations start with x=0 as an initial guess of the solution. | |
| * One can control the start using the solveWithGuess() method. | |
| * | |
| * MINRES can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink. | |
| * | |
| * \sa class ConjugateGradient, BiCGSTAB, SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner | |
| */ | |
| template< typename _MatrixType, int _UpLo, typename _Preconditioner> | |
| class MINRES : public IterativeSolverBase<MINRES<_MatrixType,_UpLo,_Preconditioner> > | |
| { | |
| typedef IterativeSolverBase<MINRES> Base; | |
| using Base::matrix; | |
| using Base::m_error; | |
| using Base::m_iterations; | |
| using Base::m_info; | |
| using Base::m_isInitialized; | |
| public: | |
| using Base::_solve_impl; | |
| typedef _MatrixType MatrixType; | |
| typedef typename MatrixType::Scalar Scalar; | |
| typedef typename MatrixType::RealScalar RealScalar; | |
| typedef _Preconditioner Preconditioner; | |
| enum {UpLo = _UpLo}; | |
| public: | |
| /** Default constructor. */ | |
| MINRES() : Base() {} | |
| /** Initialize the solver with matrix \a A for further \c Ax=b solving. | |
| * | |
| * This constructor is a shortcut for the default constructor followed | |
| * by a call to compute(). | |
| * | |
| * \warning this class stores a reference to the matrix A as well as some | |
| * precomputed values that depend on it. Therefore, if \a A is changed | |
| * this class becomes invalid. Call compute() to update it with the new | |
| * matrix A, or modify a copy of A. | |
| */ | |
| template<typename MatrixDerived> | |
| explicit MINRES(const EigenBase<MatrixDerived>& A) : Base(A.derived()) {} | |
| /** Destructor. */ | |
| ~MINRES(){} | |
| /** \internal */ | |
| template<typename Rhs,typename Dest> | |
| void _solve_vector_with_guess_impl(const Rhs& b, Dest& x) const | |
| { | |
| typedef typename Base::MatrixWrapper MatrixWrapper; | |
| typedef typename Base::ActualMatrixType ActualMatrixType; | |
| enum { | |
| TransposeInput = (!MatrixWrapper::MatrixFree) | |
| && (UpLo==(Lower|Upper)) | |
| && (!MatrixType::IsRowMajor) | |
| && (!NumTraits<Scalar>::IsComplex) | |
| }; | |
| typedef typename internal::conditional<TransposeInput,Transpose<const ActualMatrixType>, ActualMatrixType const&>::type RowMajorWrapper; | |
| EIGEN_STATIC_ASSERT(EIGEN_IMPLIES(MatrixWrapper::MatrixFree,UpLo==(Lower|Upper)),MATRIX_FREE_CONJUGATE_GRADIENT_IS_COMPATIBLE_WITH_UPPER_UNION_LOWER_MODE_ONLY); | |
| typedef typename internal::conditional<UpLo==(Lower|Upper), | |
| RowMajorWrapper, | |
| typename MatrixWrapper::template ConstSelfAdjointViewReturnType<UpLo>::Type | |
| >::type SelfAdjointWrapper; | |
| m_iterations = Base::maxIterations(); | |
| m_error = Base::m_tolerance; | |
| RowMajorWrapper row_mat(matrix()); | |
| internal::minres(SelfAdjointWrapper(row_mat), b, x, | |
| Base::m_preconditioner, m_iterations, m_error); | |
| m_info = m_error <= Base::m_tolerance ? Success : NoConvergence; | |
| } | |
| protected: | |
| }; | |
| } // end namespace Eigen | |