A GMRES solver for sparse square problems. More...

#include <GMRES.h>

Inheritance diagram for Eigen::GMRES< MatrixType_, Preconditioner_ >:

Public Types
typedef MatrixType_	MatrixType

typedef MatrixType::Scalar	Scalar

typedef MatrixType::RealScalar	RealScalar

typedef Preconditioner_	Preconditioner

Public Types inherited from Eigen::IterativeSolverBase< GMRES< MatrixType_, Preconditioner_ > >
enum

typedef internal::traits< GMRES< MatrixType_, Preconditioner_ > >::MatrixType	MatrixType

typedef internal::traits< GMRES< MatrixType_, Preconditioner_ > >::Preconditioner	Preconditioner

typedef MatrixType::Scalar	Scalar

typedef MatrixType::StorageIndex	StorageIndex

typedef MatrixType::RealScalar	RealScalar

Public Member Functions
	GMRES ()

template<typename MatrixDerived >
	GMRES (const EigenBase< MatrixDerived > &A)

	~GMRES ()

Index	get_restart ()

void	set_restart (const Index restart)

template<typename Rhs , typename Dest >
void	_solve_vector_with_guess_impl (const Rhs &b, Dest &x) const

template<typename Rhs , typename Dest >
void	_solve_impl (const Rhs &b, Dest &x) const

Public Member Functions inherited from Eigen::IterativeSolverBase< GMRES< MatrixType_, Preconditioner_ > >
	IterativeSolverBase ()

	IterativeSolverBase (const EigenBase< MatrixDerived > &A)

	IterativeSolverBase (IterativeSolverBase &&)=default

	~IterativeSolverBase ()

GMRES< MatrixType_, Preconditioner_ > &	analyzePattern (const EigenBase< MatrixDerived > &A)

GMRES< MatrixType_, Preconditioner_ > &	factorize (const EigenBase< MatrixDerived > &A)

GMRES< MatrixType_, Preconditioner_ > &	compute (const EigenBase< MatrixDerived > &A)

EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

RealScalar	tolerance () const

GMRES< MatrixType_, Preconditioner_ > &	setTolerance (const RealScalar &tolerance)

Preconditioner &	preconditioner ()

const Preconditioner &	preconditioner () const

Index	maxIterations () const

GMRES< MatrixType_, Preconditioner_ > &	setMaxIterations (Index maxIters)

Index	iterations () const

RealScalar	error () const

const SolveWithGuess< GMRES< MatrixType_, Preconditioner_ >, Rhs, Guess >	solveWithGuess (const MatrixBase< Rhs > &b, const Guess &x0) const

ComputationInfo	info () const

void	_solve_with_guess_impl (const Rhs &b, SparseMatrixBase< DestDerived > &aDest) const

std::enable_if_t< Rhs::ColsAtCompileTime !=1 &&DestDerived::ColsAtCompileTime !=1 >	_solve_with_guess_impl (const Rhs &b, MatrixBase< DestDerived > &aDest) const

std::enable_if_t< Rhs::ColsAtCompileTime==1\|\|DestDerived::ColsAtCompileTime==1 >	_solve_with_guess_impl (const Rhs &b, MatrixBase< DestDerived > &dest) const

void	_solve_impl (const Rhs &b, Dest &x) const

GMRES< MatrixType_, Preconditioner_ > &	derived ()

const GMRES< MatrixType_, Preconditioner_ > &	derived () const

Public Member Functions inherited from Eigen::SparseSolverBase< Derived >
	SparseSolverBase ()

	SparseSolverBase (SparseSolverBase &&other)

	~SparseSolverBase ()

Derived &	derived ()

const Derived &	derived () const

template<typename Rhs >
const Solve< Derived, Rhs >	solve (const MatrixBase< Rhs > &b) const

template<typename Rhs >
const Solve< Derived, Rhs >	solve (const SparseMatrixBase< Rhs > &b) const

template<typename Rhs , typename Dest >
void	_solve_impl (const SparseMatrixBase< Rhs > &b, SparseMatrixBase< Dest > &dest) const

Private Types
typedef IterativeSolverBase< GMRES >	Base

Private Member Functions
const ActualMatrixType &	matrix () const

Private Attributes
Index	m_restart

RealScalar	m_error

ComputationInfo	m_info

bool	m_isInitialized

Index	m_iterations

Additional Inherited Members
Protected Types inherited from Eigen::IterativeSolverBase< GMRES< MatrixType_, Preconditioner_ > >
typedef SparseSolverBase< GMRES< MatrixType_, Preconditioner_ > >	Base

typedef internal::generic_matrix_wrapper< MatrixType >	MatrixWrapper

typedef MatrixWrapper::ActualMatrixType	ActualMatrixType

Protected Member Functions inherited from Eigen::IterativeSolverBase< GMRES< MatrixType_, Preconditioner_ > >
void	init ()

const ActualMatrixType &	matrix () const

void	grab (const InputType &A)

Protected Attributes inherited from Eigen::IterativeSolverBase< GMRES< MatrixType_, Preconditioner_ > >
MatrixWrapper	m_matrixWrapper

Preconditioner	m_preconditioner

Index	m_maxIterations

RealScalar	m_tolerance

RealScalar	m_error

Index	m_iterations

ComputationInfo	m_info

bool	m_analysisIsOk

bool	m_factorizationIsOk

bool	m_isInitialized

Protected Attributes inherited from Eigen::SparseSolverBase< Derived >
bool	m_isInitialized

Detailed Description

template<typename MatrixType_, typename Preconditioner_>
class Eigen::GMRES< MatrixType_, Preconditioner_ >

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.

Template Parameters

MatrixType_	the type of the sparse matrix A, can be a dense or a sparse matrix.
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:

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);

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 example .

See also: class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner

Member Typedef Documentation

◆ Base

template<typename MatrixType_ , typename Preconditioner_ >

typedef IterativeSolverBase<GMRES> Eigen::GMRES< MatrixType_, Preconditioner_ >::Base

private

◆ MatrixType

template<typename MatrixType_ , typename Preconditioner_ >

typedef MatrixType_ Eigen::GMRES< MatrixType_, Preconditioner_ >::MatrixType

◆ Preconditioner

template<typename MatrixType_ , typename Preconditioner_ >

typedef Preconditioner_ Eigen::GMRES< MatrixType_, Preconditioner_ >::Preconditioner

◆ RealScalar

template<typename MatrixType_ , typename Preconditioner_ >

typedef MatrixType::RealScalar Eigen::GMRES< MatrixType_, Preconditioner_ >::RealScalar

◆ Scalar

template<typename MatrixType_ , typename Preconditioner_ >

typedef MatrixType::Scalar Eigen::GMRES< MatrixType_, Preconditioner_ >::Scalar

Constructor & Destructor Documentation

◆ GMRES() [1/2]

template<typename MatrixType_ , typename Preconditioner_ >

Eigen::GMRES< MatrixType_, Preconditioner_ >::GMRES ( )

inline

Default constructor.

275 : Base(), m_restart(30) {}

Eigen::GMRES::Base

IterativeSolverBase< GMRES > Base

Definition: GMRES.h:256

Eigen::GMRES::m_restart

Index m_restart

Definition: GMRES.h:264

◆ GMRES() [2/2]

template<typename MatrixType_ , typename Preconditioner_ >

template<typename MatrixDerived >

Eigen::GMRES< MatrixType_, Preconditioner_ >::GMRES ( const EigenBase< MatrixDerived > & A )

inlineexplicit

Initialize the solver with matrix A for further 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 is changed this class becomes invalid. Call compute() to update it with the new matrix A, or modify a copy of A.

288 : Base(A.derived()), m_restart(30) {}

◆ ~GMRES()

template<typename MatrixType_ , typename Preconditioner_ >

Eigen::GMRES< MatrixType_, Preconditioner_ >::~GMRES ( )

inline

290 {}

Member Function Documentation

◆ _solve_impl()

template<typename MatrixType_ , typename Preconditioner_ >

template<typename Rhs , typename Dest >

void Eigen::IterativeSolverBase< Derived >::_solve_impl	(	typename Rhs	,
		typename Dest
	)

inline

default initial guess = 0

                                                 {
     x.setZero();
     derived()._solve_with_guess_impl(b, x);
   }

◆ _solve_vector_with_guess_impl()

template<typename MatrixType_ , typename Preconditioner_ >

template<typename Rhs , typename Dest >

void Eigen::GMRES< MatrixType_, Preconditioner_ >::_solve_vector_with_guess_impl	(	const Rhs &	b,
		Dest &	x
	)		const

inline

                                                                   {
     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;
   }