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// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2011 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2012, 2014 Kolja Brix <brix@igpm.rwth-aaachen.de>
//
// 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_GMRES_H
#define EIGEN_GMRES_H
namespace Eigen {
namespace internal {
/**
* Generalized Minimal Residual Algorithm based on the
* Arnoldi algorithm implemented with Householder reflections.
*
* Parameters:
* \param mat matrix of linear system of equations
* \param rhs right hand side vector of linear system of equations
* \param x on input: initial guess, on output: solution
* \param precond preconditioner used
* \param iters on input: maximum number of iterations to perform
* on output: number of iterations performed
* \param restart number of iterations for a restart
* \param tol_error on input: relative residual tolerance
* on output: residuum achieved
*
* \sa IterativeMethods::bicgstab()
*
*
* For references, please see:
*
* Saad, Y. and Schultz, M. H.
* GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems.
* SIAM J.Sci.Stat.Comp. 7, 1986, pp. 856 - 869.
*
* Saad, Y.
* Iterative Methods for Sparse Linear Systems.
* Society for Industrial and Applied Mathematics, Philadelphia, 2003.
*
* Walker, H. F.
* Implementations of the GMRES method.
* Comput.Phys.Comm. 53, 1989, pp. 311 - 320.
*
* Walker, H. F.
* Implementation of the GMRES Method using Householder Transformations.
* SIAM J.Sci.Stat.Comp. 9, 1988, pp. 152 - 163.
*
*/
template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
bool gmres(const MatrixType & mat, const Rhs & rhs, Dest & x, const Preconditioner & precond,
Index &iters, const Index &restart, typename Dest::RealScalar & tol_error) {
using std::sqrt;
using std::abs;
typedef typename Dest::RealScalar RealScalar;
typedef typename Dest::Scalar Scalar;
typedef Matrix < Scalar, Dynamic, 1 > VectorType;
typedef Matrix < Scalar, Dynamic, Dynamic, ColMajor> FMatrixType;
const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
if(rhs.norm() <= considerAsZero)
{
x.setZero();
tol_error = 0;
return true;
}
RealScalar tol = tol_error;
const Index maxIters = iters;
iters = 0;
const Index m = mat.rows();
// residual and preconditioned residual
VectorType p0 = rhs - mat*x;
VectorType r0 = precond.solve(p0);
const RealScalar r0Norm = r0.norm();
// is initial guess already good enough?
if(r0Norm == 0)
{
tol_error = 0;
return true;
}
// storage for Hessenberg matrix and Householder data
FMatrixType H = FMatrixType::Zero(m, restart + 1);
VectorType w = VectorType::Zero(restart + 1);
VectorType tau = VectorType::Zero(restart + 1);
// storage for Jacobi rotations
std::vector < JacobiRotation < Scalar > > G(restart);
// storage for temporaries
VectorType t(m), v(m), workspace(m), x_new(m);
// generate first Householder vector
Ref<VectorType> H0_tail = H.col(0).tail(m - 1);
RealScalar beta;
r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta);
w(0) = Scalar(beta);
for (Index k = 1; k <= restart; ++k)
{
++iters;
v = VectorType::Unit(m, k - 1);
// apply Householder reflections H_{1} ... H_{k-1} to v
// TODO: use a HouseholderSequence
for (Index i = k - 1; i >= 0; --i) {
v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
}
// apply matrix M to v: v = mat * v;
t.noalias() = mat * v;
v = precond.solve(t);
// apply Householder reflections H_{k-1} ... H_{1} to v
// TODO: use a HouseholderSequence
for (Index i = 0; i < k; ++i) {
v.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
}
if (v.tail(m - k).norm() != 0.0)
{
if (k <= restart)
{
// generate new Householder vector
Ref<VectorType> Hk_tail = H.col(k).tail(m - k - 1);
v.tail(m - k).makeHouseholder(Hk_tail, tau.coeffRef(k), beta);
// apply Householder reflection H_{k} to v
v.tail(m - k).applyHouseholderOnTheLeft(Hk_tail, tau.coeffRef(k), workspace.data());
}
}
if (k > 1)
{
for (Index i = 0; i < k - 1; ++i)
{
// apply old Givens rotations to v
v.applyOnTheLeft(i, i + 1, G[i].adjoint());
}
}
if (k<m && v(k) != (Scalar) 0)
{
// determine next Givens rotation
G[k - 1].makeGivens(v(k - 1), v(k));
// apply Givens rotation to v and w
v.applyOnTheLeft(k - 1, k, G[k - 1].adjoint());
w.applyOnTheLeft(k - 1, k, G[k - 1].adjoint());
}
// insert coefficients into upper matrix triangle
H.col(k-1).head(k) = v.head(k);
tol_error = abs(w(k)) / r0Norm;
bool stop = (k==m || tol_error < tol || iters == maxIters);
if (stop || k == restart)
{
// solve upper triangular system
Ref<VectorType> y = w.head(k);
H.topLeftCorner(k, k).template triangularView <Upper>().solveInPlace(y);
// use Horner-like scheme to calculate solution vector
x_new.setZero();
for (Index i = k - 1; i >= 0; --i)
{
x_new(i) += y(i);
// apply Householder reflection H_{i} to x_new
x_new.tail(m - i).applyHouseholderOnTheLeft(H.col(i).tail(m - i - 1), tau.coeffRef(i), workspace.data());
}
x += x_new;
if(stop)
{
return true;
}
else
{
k=0;
// reset data for restart
p0.noalias() = rhs - mat*x;
r0 = precond.solve(p0);
// clear Hessenberg matrix and Householder data
H.setZero();
w.setZero();
tau.setZero();
// generate first Householder vector
r0.makeHouseholder(H0_tail, tau.coeffRef(0), beta);
w(0) = Scalar(beta);
}
}
}
return false;
}
}
template< typename _MatrixType,
typename _Preconditioner = DiagonalPreconditioner<typename _MatrixType::Scalar> >
class GMRES;
namespace internal {
template< typename _MatrixType, typename _Preconditioner>
struct traits<GMRES<_MatrixType,_Preconditioner> >
{
typedef _MatrixType MatrixType;
typedef _Preconditioner Preconditioner;
};
}
/** \ingroup IterativeLinearSolvers_Module
* \brief A GMRES solver for sparse square problems
*
* This class allows to solve for A.x = b sparse linear problems using a generalized minimal
* residual method. 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 _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
* GMRES<SparseMatrix<double> > solver(A);
* x = solver.solve(b);
* std::cout << "#iterations: " << solver.iterations() << std::endl;
* std::cout << "estimated error: " << solver.error() << std::endl;
* // update b, and solve again
* x = solver.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.
*
* GMRES can also be used in a matrix-free context, see the following \link MatrixfreeSolverExample example \endlink.
*
* \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
*/
template< typename _MatrixType, typename _Preconditioner>
class GMRES : public IterativeSolverBase<GMRES<_MatrixType,_Preconditioner> >
{
typedef IterativeSolverBase<GMRES> Base;
using Base::matrix;
using Base::m_error;
using Base::m_iterations;
using Base::m_info;
using Base::m_isInitialized;
private:
Index m_restart;
public:
using Base::_solve_impl;
typedef _MatrixType MatrixType;
typedef typename MatrixType::Scalar Scalar;
typedef typename MatrixType::RealScalar RealScalar;
typedef _Preconditioner Preconditioner;
public:
/** Default constructor. */
GMRES() : Base(), m_restart(30) {}
/** 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 GMRES(const EigenBase<MatrixDerived>& A) : Base(A.derived()), m_restart(30) {}
~GMRES() {}
/** Get the number of iterations after that a restart is performed.
*/
Index get_restart() { return m_restart; }
/** Set the number of iterations after that a restart is performed.
* \param restart number of iterations for a restarti, default is 30.
*/
void set_restart(const Index restart) { m_restart=restart; }
/** \internal */
template<typename Rhs,typename Dest>
void _solve_vector_with_guess_impl(const Rhs& b, Dest& x) const
{
m_iterations = Base::maxIterations();
m_error = Base::m_tolerance;
bool ret = internal::gmres(matrix(), b, x, Base::m_preconditioner, m_iterations, m_restart, m_error);
m_info = (!ret) ? NumericalIssue
: m_error <= Base::m_tolerance ? Success
: NoConvergence;
}
protected:
};
} // end namespace Eigen
#endif // EIGEN_GMRES_H