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345 lines
14 KiB
C++
345 lines
14 KiB
C++
// 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 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 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, int& iters,
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typename Dest::RealScalar& tol_error)
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{
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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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// initialize
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const int maxIters(iters); // initialize maxIters to iters
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const int N(mat.cols()); // the size of the matrix
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const RealScalar rhsNorm2(rhs.squaredNorm());
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// const RealScalar threshold(tol_error); // threshold for original convergence criterion, see below
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const RealScalar threshold2(tol_error*tol_error*rhsNorm2); // convergence threshold
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// VectorType v(VectorType::Zero(N));
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// VectorType v_hat(rhs-mat*x);
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// Compute initial residual
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VectorType residual(rhs-mat*x);
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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 = residual; //initialize v_new
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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_new = sqrt(v_new.dot(w_new));
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v_new /= beta_new;
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w_new /= beta_new;
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// RealScalar beta(v_hat.norm());
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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(VectorType::Zero(N)); // initialize p_oold=0
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VectorType p_old(p_oold); // initialize p_old=0
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VectorType p(N); // will be initialized in loop
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//RealScalar eta(beta); // CHANGE THIS
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RealScalar norm_rMR=beta;
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const RealScalar norm_r0(beta);
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RealScalar eta(1.0);
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// VectorType v_old(N), Av(N), w_oold(N); // preallocate temporaty vectors used in iteration
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RealScalar residualNorm2; // not needed for original convergnce criterion
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int n = 0;
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while ( n < maxIters ){
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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 corresonds
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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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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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v = v_new; // update
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w = w_new; // update
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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_new = sqrt(v_new.dot(w_new)); // compute beta_new
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v_new /= beta_new; // overwrite v_new
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w_new /= beta_new; // overwrite w_new
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//
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//
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//
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//
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//
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//
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//
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//
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// // VectorType v_old(v); // now pre-allocated
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// v_old = v;
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// v=v_hat/beta;
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//// VectorType Av(mat*v); // now pre-allocated
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// Av = mat*v;
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// RealScalar alpha(v.transpose()*Av);
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// v_hat=Av-alpha*v-beta*v_old;
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// RealScalar beta_old(beta);
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// beta=v_hat.norm();
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// Apply QR
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// RealScalar c_oold(c_old); // store old-old cosine
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// c_old=c; // store old cosine
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// RealScalar s_oold(s_old); // store old-old sine
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// s_old=s; // store old sine
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// const RealScalar r1_hat=c_old *alpha-c_oold*s_old *beta_old;
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// const RealScalar r1 =std::pow(std::pow(r1_hat,2)+std::pow(beta,2),0.5);
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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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// Compute new Givens rotation
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const RealScalar r1_hat=c*alpha-c_old*s*beta;
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const RealScalar r1 =std::pow(std::pow(r1_hat,2)+std::pow(beta_new,2),0.5);
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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/r1; // new sine
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// update w
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// VectorType w_oold(w_old); // now pre-allocated
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p_oold = p_old;
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p_old = p;
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p=(w-r2*p_old-r3*p_oold) /r1;
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// update x
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x += c*eta*p;
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norm_rMR *= std::fabs(s);
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residualNorm2 = (mat*x-rhs).squaredNorm(); // DOES mat*x NEED TO BE RECOMPUTED ????
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//if(norm_rMR/norm_r0 < threshold){ // original convergence criterion, does not require "mat*x"
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if ( residualNorm2 < threshold2){
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break;
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}
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eta=-s*eta; // update eta
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n++; // increment iteration
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}
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tol_error = std::sqrt(residualNorm2 / rhsNorm2); // return error
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iters = n; // return number of iterations
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}
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}
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template< typename _MatrixType, int _UpLo=Lower,
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typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
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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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* or Upper. 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. Here is a step by
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* step execution example starting with a random guess and printing the evolution
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* of the estimated error:
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* * \code
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* x = VectorXd::Random(n);
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* mr.setMaxIterations(1);
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* int i = 0;
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* do {
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* x = mr.solveWithGuess(b,x);
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* std::cout << i << " : " << mr.error() << std::endl;
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* ++i;
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* } while (mr.info()!=Success && i<100);
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* \endcode
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* Note that such a step by step excution is slightly slower.
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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::mp_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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typedef _MatrixType MatrixType;
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typedef typename MatrixType::Scalar Scalar;
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typedef typename MatrixType::Index Index;
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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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MINRES(const MatrixType& A) : Base(A) {}
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/** Destructor. */
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~MINRES(){}
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/** \returns the solution x of \f$ A x = b \f$ using the current decomposition of A
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* \a x0 as an initial solution.
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*
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* \sa compute()
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*/
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template<typename Rhs,typename Guess>
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inline const internal::solve_retval_with_guess<MINRES, Rhs, Guess>
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solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const
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{
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eigen_assert(m_isInitialized && "MINRES is not initialized.");
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eigen_assert(Base::rows()==b.rows()
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&& "MINRES::solve(): invalid number of rows of the right hand side matrix b");
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return internal::solve_retval_with_guess
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<MINRES, Rhs, Guess>(*this, b.derived(), x0);
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}
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/** \internal */
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template<typename Rhs,typename Dest>
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void _solveWithGuess(const Rhs& b, Dest& x) const
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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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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(mp_matrix->template selfadjointView<UpLo>(), 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(const Rhs& b, Dest& x) const
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{
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x.setOnes();
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_solveWithGuess(b,x);
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}
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protected:
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};
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namespace internal {
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template<typename _MatrixType, int _UpLo, typename _Preconditioner, typename Rhs>
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struct solve_retval<MINRES<_MatrixType,_UpLo,_Preconditioner>, Rhs>
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: solve_retval_base<MINRES<_MatrixType,_UpLo,_Preconditioner>, Rhs>
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{
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typedef MINRES<_MatrixType,_UpLo,_Preconditioner> Dec;
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EIGEN_MAKE_SOLVE_HELPERS(Dec,Rhs)
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template<typename Dest> void evalTo(Dest& dst) const
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{
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dec()._solve(rhs(),dst);
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}
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};
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} // end namespace internal
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} // end namespace Eigen
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#endif // EIGEN_MINRES_H
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