A Restarted GMRES with deflation. More...

#include <DGMRES.h>

Inheritance diagram for Eigen::DGMRES< _MatrixType, _Preconditioner >:

Public Types
typedef _MatrixType	MatrixType

typedef MatrixType::Scalar	Scalar

typedef MatrixType::Index	Index

typedef MatrixType::StorageIndex	StorageIndex

typedef MatrixType::RealScalar	RealScalar

typedef _Preconditioner	Preconditioner

typedef Matrix< Scalar, Dynamic, Dynamic >	DenseMatrix

typedef Matrix< RealScalar, Dynamic, Dynamic >	DenseRealMatrix

typedef Matrix< Scalar, Dynamic, 1 >	DenseVector

typedef Matrix< RealScalar, Dynamic, 1 >	DenseRealVector

typedef Matrix< std::complex< RealScalar >, Dynamic, 1 >	ComplexVector

Public Types inherited from Eigen::IterativeSolverBase< DGMRES< _MatrixType, _Preconditioner > >
enum

typedef internal::traits< DGMRES< _MatrixType, _Preconditioner > >::MatrixType	MatrixType

typedef internal::traits< DGMRES< _MatrixType, _Preconditioner > >::Preconditioner	Preconditioner

typedef MatrixType::Scalar	Scalar

typedef MatrixType::StorageIndex	StorageIndex

typedef MatrixType::RealScalar	RealScalar

Public Member Functions
	DGMRES ()
	Default constructor. More...

template<typename MatrixDerived >
	DGMRES (const EigenBase< MatrixDerived > &A)
	Initialize the solver with matrix A for further `Ax=b` solving. More...

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

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

int	restart ()
	Get the restart value.

void	set_restart (const int restart)
	Set the restart value (default is 30)

void	setEigenv (const int neig)
	Set the number of eigenvalues to deflate at each restart.

int	deflSize ()
	Get the size of the deflation subspace size.

void	setMaxEigenv (const int maxNeig)
	Set the maximum size of the deflation subspace.

Public Member Functions inherited from Eigen::IterativeSolverBase< DGMRES< _MatrixType, _Preconditioner > >
	IterativeSolverBase ()
	Default constructor. More...

	IterativeSolverBase (const EigenBase< MatrixDerived > &A)
	Initialize the solver with matrix A for further `Ax=b` solving. More...

DGMRES< _MatrixType, _Preconditioner > &	analyzePattern (const EigenBase< MatrixDerived > &A)
	Initializes the iterative solver for the sparsity pattern of the matrix A for further solving `Ax=b` problems. More...

DGMRES< _MatrixType, _Preconditioner > &	factorize (const EigenBase< MatrixDerived > &A)
	Initializes the iterative solver with the numerical values of the matrix A for further solving `Ax=b` problems. More...

DGMRES< _MatrixType, _Preconditioner > &	compute (const EigenBase< MatrixDerived > &A)
	Initializes the iterative solver with the matrix A for further solving `Ax=b` problems. More...

Index	rows () const

Index	cols () const

RealScalar	tolerance () const

DGMRES< _MatrixType, _Preconditioner > &	setTolerance (const RealScalar &tolerance)
	Sets the tolerance threshold used by the stopping criteria. More...

Preconditioner &	preconditioner ()

const Preconditioner &	preconditioner () const

Index	maxIterations () const

DGMRES< _MatrixType, _Preconditioner > &	setMaxIterations (Index maxIters)
	Sets the max number of iterations. More...

Index	iterations () const

RealScalar	error () const

const SolveWithGuess< DGMRES< _MatrixType, _Preconditioner >, Rhs, Guess >	solveWithGuess (const MatrixBase< Rhs > &b, const Guess &x0) const

ComputationInfo	info () const

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

Public Member Functions inherited from Eigen::SparseSolverBase< DGMRES< _MatrixType, _Preconditioner > >
	SparseSolverBase ()
	Default constructor.

DGMRES< _MatrixType, _Preconditioner > &	derived ()

const DGMRES< _MatrixType, _Preconditioner > &	derived () const

const Solve< DGMRES< _MatrixType, _Preconditioner >, Rhs >	solve (const MatrixBase< Rhs > &b) const

const Solve< DGMRES< _MatrixType, _Preconditioner >, Rhs >	solve (const SparseMatrixBase< Rhs > &b) const

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

Protected Member Functions
template<typename Rhs , typename Dest >
void	dgmres (const MatrixType &mat, const Rhs &rhs, Dest &x, const Preconditioner &precond) const
	Perform several cycles of restarted GMRES with modified Gram Schmidt,. More...

template<typename Dest >
int	dgmresCycle (const MatrixType &mat, const Preconditioner &precond, Dest &x, DenseVector &r0, RealScalar &beta, const RealScalar &normRhs, int &nbIts) const
	Perform one restart cycle of DGMRES. More...

int	dgmresComputeDeflationData (const MatrixType &mat, const Preconditioner &precond, const Index &it, StorageIndex &neig) const

template<typename RhsType , typename DestType >
int	dgmresApplyDeflation (const RhsType &In, DestType &Out) const

ComplexVector	schurValues (const ComplexSchur< DenseMatrix > &schurofH) const

ComplexVector	schurValues (const RealSchur< DenseMatrix > &schurofH) const

void	dgmresInitDeflation (Index &rows) const

Protected Member Functions inherited from Eigen::IterativeSolverBase< DGMRES< _MatrixType, _Preconditioner > >
void	init ()

const ActualMatrixType &	matrix () const

void	grab (const InputType &A)

Protected Attributes
DenseMatrix	m_V

DenseMatrix	m_H

DenseMatrix	m_Hes

Index	m_restart

DenseMatrix	m_U

DenseMatrix	m_MU

DenseMatrix	m_T

PartialPivLU< DenseMatrix >	m_luT

StorageIndex	m_neig

int	m_r

int	m_maxNeig

RealScalar	m_lambdaN

bool	m_isDeflAllocated

bool	m_isDeflInitialized

RealScalar	m_smv

bool	m_force

Protected Attributes inherited from Eigen::IterativeSolverBase< DGMRES< _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

Protected Attributes inherited from Eigen::SparseSolverBase< DGMRES< _MatrixType, _Preconditioner > >
bool	m_isInitialized

Additional Inherited Members
Protected Types inherited from Eigen::IterativeSolverBase< DGMRES< _MatrixType, _Preconditioner > >
typedef SparseSolverBase< DGMRES< _MatrixType, _Preconditioner > >	Base

typedef internal::generic_matrix_wrapper< MatrixType >	MatrixWrapper

typedef MatrixWrapper::ActualMatrixType	ActualMatrixType

Detailed Description

template<typename _MatrixType, typename _Preconditioner>
class Eigen::DGMRES< _MatrixType, _Preconditioner >

A Restarted GMRES with deflation.

This class implements a modification of the GMRES solver for sparse linear systems. The basis is built with modified Gram-Schmidt. At each restart, a few approximated eigenvectors corresponding to the smallest eigenvalues are used to build a preconditioner for the next cycle. This preconditioner for deflation can be combined with any other preconditioner, the IncompleteLUT for instance. The preconditioner is applied at right of the matrix and the combination is multiplicative.

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 Typical usage : SparseMatrix<double> A; VectorXd x, b; //Fill A and b ... DGMRES<SparseMatrix<double> > solver; solver.set_restart(30); // Set restarting value solver.setEigenv(1); // Set the number of eigenvalues to deflate solver.compute(A); x = solver.solve(b);

DGMRES can also be used in a matrix-free context, see the following example .

References : [1] D. NUENTSA WAKAM and F. PACULL, Memory Efficient Hybrid Algebraic Solvers for Linear Systems Arising from Compressible Flows, Computers and Fluids, In Press, http://dx.doi.org/10.1016/j.compfluid.2012.03.023 [2] K. Burrage and J. Erhel, On the performance of various adaptive preconditioned GMRES strategies, 5(1998), 101-121. [3] J. Erhel, K. Burrage and B. Pohl, Restarted GMRES preconditioned by deflation,J. Computational and Applied Mathematics, 69(1996), 303-318.

Constructor & Destructor Documentation

§ DGMRES() [1/2]

template<typename _MatrixType, typename _Preconditioner>

Eigen::DGMRES< _MatrixType, _Preconditioner >::DGMRES ( )

inline

Default constructor.

§ DGMRES() [2/2]

template<typename _MatrixType, typename _Preconditioner>

template<typename MatrixDerived >

Eigen::DGMRES< _MatrixType, _Preconditioner >::DGMRES ( 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.

Member Function Documentation

§ dgmres()

template<typename _MatrixType , typename _Preconditioner >

template<typename Rhs , typename Dest >

void Eigen::DGMRES< _MatrixType, _Preconditioner >::dgmres	(	const MatrixType &	mat,
		const Rhs &	rhs,
		Dest &	x,
		const Preconditioner &	precond
	)		const

protected

Perform several cycles of restarted GMRES with modified Gram Schmidt,.

A right preconditioner is used combined with deflation.

§ dgmresCycle()

template<typename _MatrixType , typename _Preconditioner >

template<typename Dest >

int Eigen::DGMRES< _MatrixType, _Preconditioner >::dgmresCycle	(	const MatrixType &	mat,
		const Preconditioner &	precond,
		Dest &	x,
		DenseVector &	r0,
		RealScalar &	beta,
		const RealScalar &	normRhs,
		int &	nbIts
	)		const

protected

Perform one restart cycle of DGMRES.

Parameters

mat	The coefficient matrix
precond	The preconditioner
x	the new approximated solution
r0	The initial residual vector
beta	The norm of the residual computed so far
normRhs	The norm of the right hand side vector
nbIts	The number of iterations

The documentation for this class was generated from the following file:

third-party-libs/eigen3/unsupported/Eigen/src/IterativeSolvers/DGMRES.h

Public Types

Public Member Functions

Protected Member Functions

Protected Attributes

Additional Inherited Members