// 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: 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,
int &iters, const int &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 > FMatrixType;
RealScalar tol = tol_error;
const int maxIters = iters;
iters = 0;
const int m = mat.rows();
VectorType p0 = rhs - mat*x;
VectorType r0 = precond.solve(p0);
// is initial guess already good enough?
if(abs(r0.norm()) < tol) {
return true;
}
VectorType w = VectorType::Zero(restart + 1);
FMatrixType H = FMatrixType::Zero(m, restart + 1); // Hessenberg matrix
VectorType tau = VectorType::Zero(restart + 1);
std::vector < JacobiRotation < Scalar > > G(restart);
// generate first Householder vector
VectorType e(m-1);
RealScalar beta;
r0.makeHouseholder(e, tau.coeffRef(0), beta);
w(0)=(Scalar) beta;
H.bottomLeftCorner(m - 1, 1) = e;
for (int k = 1; k <= restart; ++k) {
++iters;
VectorType v = VectorType::Unit(m, k - 1), workspace(m);
// apply Householder reflections H_{1} ... H_{k-1} to v
for (int 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;
VectorType t=mat*v;
v=precond.solve(t);
// apply Householder reflections H_{k-1} ... H_{1} to v
for (int 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
VectorType e(m - k - 1);
RealScalar beta;
v.tail(m - k).makeHouseholder(e, tau.coeffRef(k), beta);
H.col(k).tail(m - k - 1) = e;
// apply Householder reflection H_{k} to v
v.tail(m - k).applyHouseholderOnTheLeft(H.col(k).tail(m - k - 1), tau.coeffRef(k), workspace.data());
}
}
if (k > 1) {
for (int 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);
bool stop=(k==m || abs(w(k)) < tol || iters == maxIters);
if (stop || k == restart) {
// solve upper triangular system
VectorType y = w.head(k);
H.topLeftCorner(k, k).template triangularView < Eigen::Upper > ().solveInPlace(y);
// use Horner-like scheme to calculate solution vector
VectorType x_new = y(k - 1) * VectorType::Unit(m, k - 1);
// apply Householder reflection H_{k} to x_new
x_new.tail(m - k + 1).applyHouseholderOnTheLeft(H.col(k - 1).tail(m - k), tau.coeffRef(k - 1), workspace.data());
for (int i = k - 2; i >= 0; --i) {
x_new += y(i) * VectorType::Unit(m, 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 a restart r0 = rhs - mat * x;
VectorType p0=mat*x;
VectorType p1=precond.solve(p0);
r0 = rhs - p1;
// r0_sqnorm = r0.squaredNorm();
w = VectorType::Zero(restart + 1);
H = FMatrixType::Zero(m, restart + 1);
tau = VectorType::Zero(restart + 1);
// generate first Householder vector
RealScalar beta;
r0.makeHouseholder(e, tau.coeffRef(0), beta);
w(0)=(Scalar) beta;
H.bottomLeftCorner(m - 1, 1) = e;
}
}
}
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. 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);
* solver.setMaxIterations(1);
* int i = 0;
* do {
* x = solver.solveWithGuess(b,x);
* std::cout << i << " : " << solver.error() << std::endl;
* ++i;
* } while (solver.info()!=Success && i<100);
* \endcode
* Note that such a step by step excution is slightly slower.
*
* \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
*/
template< typename _MatrixType, typename _Preconditioner>
class GMRES : public IterativeSolverBase<GMRES<_MatrixType,_Preconditioner> >
{
typedef IterativeSolverBase<GMRES> Base;
using Base::mp_matrix;
using Base::m_error;
using Base::m_iterations;
using Base::m_info;
using Base::m_isInitialized;
private:
int m_restart;
public:
typedef _MatrixType MatrixType;
typedef typename MatrixType::Scalar Scalar;
typedef typename MatrixType::Index Index;
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.
*/
GMRES(const MatrixType& A) : Base(A), m_restart(30) {}
~GMRES() {}
/** Get the number of iterations after that a restart is performed.
*/
int 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 int restart) { m_restart=restart; }
/** \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<GMRES, Rhs, Guess>
solveWithGuess(const MatrixBase<Rhs>& b, const Guess& x0) const
{
eigen_assert(m_isInitialized && "GMRES is not initialized.");
eigen_assert(Base::rows()==b.rows()
&& "GMRES::solve(): invalid number of rows of the right hand side matrix b");
return internal::solve_retval_with_guess
<GMRES, Rhs, Guess>(*this, b.derived(), x0);
}
/** \internal */
template<typename Rhs,typename Dest>
void _solveWithGuess(const Rhs& b, Dest& x) const
{
bool failed = false;
for(int j=0; j<b.cols(); ++j)
{
m_iterations = Base::maxIterations();
m_error = Base::m_tolerance;
typename Dest::ColXpr xj(x,j);
if(!internal::gmres(*mp_matrix, b.col(j), xj, Base::m_preconditioner, m_iterations, m_restart, m_error))
failed = true;
}
m_info = failed ? NumericalIssue
: m_error <= Base::m_tolerance ? Success
: NoConvergence;
m_isInitialized = true;
}
/** \internal */
template<typename Rhs,typename Dest>
void _solve(const Rhs& b, Dest& x) const
{
x = b;
if(x.squaredNorm() == 0) return; // Check Zero right hand side
_solveWithGuess(b,x);
}
protected:
};
namespace internal {
template<typename _MatrixType, typename _Preconditioner, typename Rhs>
struct solve_retval<GMRES<_MatrixType, _Preconditioner>, Rhs>
: solve_retval_base<GMRES<_MatrixType, _Preconditioner>, Rhs>
{
typedef GMRES<_MatrixType, _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_GMRES_H