| // 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 Gael Guennebaud <gael.guennebaud@inria.fr> |
| // |
| // 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/. |
| |
| |
| #ifndef EIGEN_MINRES_H_ |
| #define EIGEN_MINRES_H_ |
| |
| |
| 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, int& 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; |
| |
| // initialize |
| const int maxIters(iters); // initialize maxIters to iters |
| const int N(mat.cols()); // the size of the matrix |
| const RealScalar rhsNorm2(rhs.squaredNorm()); |
| 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 && "PRECONDITIONER IS NOT POSITIVE DEFINITE"); |
| RealScalar beta_new(sqrt(beta_new2)); |
| const RealScalar beta_one(beta_new); |
| v_new /= beta_new; |
| w_new /= 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 |
| const VectorType v_old(v); // NOT SURE IF CREATING v_old EVERY ITERATION IS EFFICIENT |
| v = v_new; // update |
| // w = w_new; // update |
| const VectorType w(w_new); // NOT SURE IF CREATING w EVERY ITERATION IS EFFICIENT |
| 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 && "PRECONDITIONER IS NOT POSITIVE DEFINITE"); |
| beta_new = sqrt(beta_new2); // compute beta_new |
| v_new /= beta_new; // overwrite v_new for next iteration |
| w_new /= beta_new; // overwrite w_new for next iteration |
| |
| // 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; |
| const VectorType p_oold(p_old); // NOT SURE IF CREATING p_oold EVERY ITERATION IS EFFICIENT |
| p_old = p; |
| p.noalias()=(w-r2*p_old-r3*p_oold) /r1; // IS NOALIAS REQUIRED? |
| x += beta_one*c*eta*p; |
| residualNorm2 *= s*s; |
| |
| if ( residualNorm2 < threshold2){ |
| break; |
| } |
| |
| eta=-s*eta; // update eta |
| iters++; // increment iteration number (for output purposes) |
| } |
| tol_error = std::sqrt(residualNorm2 / rhsNorm2); // return error. Note that this is the estimated error. The real error |Ax-b|/|b| may be slightly larger |
| } |
| |
| } |
| |
| template< typename _MatrixType, int _UpLo=Lower, |
| typename _Preconditioner = IdentityPreconditioner> |
| // typename _Preconditioner = IdentityPreconditioner<typename _MatrixType::Scalar> > // preconditioner must be positive definite |
| 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 |
| * or Upper. 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. Here is a step by |
| * step execution example starting with a random guess and printing the evolution |
| * of the estimated error: |
| * * \code |
| * x = VectorXd::Random(n); |
| * mr.setMaxIterations(1); |
| * int i = 0; |
| * do { |
| * x = mr.solveWithGuess(b,x); |
| * std::cout << i << " : " << mr.error() << std::endl; |
| * ++i; |
| * } while (mr.info()!=Success && i<100); |
| * \endcode |
| * Note that such a step by step excution is slightly slower. |
| * |
| * \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::mp_matrix; |
| using Base::m_error; |
| using Base::m_iterations; |
| using Base::m_info; |
| using Base::m_isInitialized; |
| public: |
| typedef _MatrixType MatrixType; |
| typedef typename MatrixType::Scalar Scalar; |
| typedef typename MatrixType::Index Index; |
| 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. |
| */ |
| MINRES(const MatrixType& A) : Base(A) {} |
| |
| /** Destructor. */ |
| ~MINRES(){} |
| |
| /** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A |
| * \a x0 as an initial solution. |
| * |
| * \sa compute() |
| */ |
| template<typename Rhs,typename Guess> |
| inline const internal::solve_retval_with_guess<MINRES, Rhs, Guess> |
| solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const |
| { |
| eigen_assert(m_isInitialized && "MINRES is not initialized."); |
| eigen_assert(Base::rows()==b.rows() |
| && "MINRES::solve(): invalid number of rows of the right hand side matrix b"); |
| return internal::solve_retval_with_guess |
| <MINRES, Rhs, Guess>(*this, b.derived(), x0); |
| } |
| |
| /** \internal */ |
| template<typename Rhs,typename Dest> |
| void _solveWithGuess(const Rhs& b, Dest& x) const |
| { |
| m_iterations = Base::maxIterations(); |
| m_error = Base::m_tolerance; |
| |
| for(int j=0; j<b.cols(); ++j) |
| { |
| m_iterations = Base::maxIterations(); |
| m_error = Base::m_tolerance; |
| |
| typename Dest::ColXpr xj(x,j); |
| internal::minres(mp_matrix->template selfadjointView<UpLo>(), b.col(j), xj, |
| Base::m_preconditioner, m_iterations, m_error); |
| } |
| |
| m_isInitialized = true; |
| m_info = m_error <= Base::m_tolerance ? Success : NoConvergence; |
| } |
| |
| /** \internal */ |
| template<typename Rhs,typename Dest> |
| void _solve(const Rhs& b, Dest& x) const |
| { |
| x.setZero(); |
| _solveWithGuess(b,x); |
| } |
| |
| protected: |
| |
| }; |
| |
| namespace internal { |
| |
| template<typename _MatrixType, int _UpLo, typename _Preconditioner, typename Rhs> |
| struct solve_retval<MINRES<_MatrixType,_UpLo,_Preconditioner>, Rhs> |
| : solve_retval_base<MINRES<_MatrixType,_UpLo,_Preconditioner>, Rhs> |
| { |
| typedef MINRES<_MatrixType,_UpLo,_Preconditioner> Dec; |
| EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs) |
| |
| template<typename Dest> void evalTo(Dest& dst) const |
| { |
| dec()._solve(rhs(),dst); |
| } |
| }; |
| |
| } // end namespace internal |
| |
| } // end namespace Eigen |
| |
| #endif // EIGEN_MINRES_H |
| |