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. More...

#include <DGMRES.h>

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

Public Types
typedef _MatrixType	MatrixType

typedef MatrixType::Scalar	Scalar

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 ()

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

	~DGMRES ()

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

Index	restart ()

void	set_restart (const Index restart)

void	setEigenv (const Index neig)

Index	deflSize ()

void	setMaxEigenv (const Index maxNeig)

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

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

template<typename Rhs , typename DestDerived >
internal::enable_if< Rhs::ColsAtCompileTime!=1 &&DestDerived::ColsAtCompileTime!=1 >::type	_solve_with_guess_impl (const Rhs &b, MatrixBase< DestDerived > &aDest) const

template<typename Rhs , typename DestDerived >
internal::enable_if< Rhs::ColsAtCompileTime==1\|\|DestDerived::ColsAtCompileTime==1 >::type	_solve_with_guess_impl (const Rhs &b, MatrixBase< DestDerived > &dest) const

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

	IterativeSolverBase (const EigenBase< MatrixDerived > &A)

	~IterativeSolverBase ()

DGMRES< _MatrixType, _Preconditioner > &	analyzePattern (const EigenBase< MatrixDerived > &A)

DGMRES< _MatrixType, _Preconditioner > &	factorize (const EigenBase< MatrixDerived > &A)

DGMRES< _MatrixType, _Preconditioner > &	compute (const EigenBase< MatrixDerived > &A)

EIGEN_CONSTEXPR Index	rows () const EIGEN_NOEXCEPT

EIGEN_CONSTEXPR Index	cols () const EIGEN_NOEXCEPT

RealScalar	tolerance () const

DGMRES< _MatrixType, _Preconditioner > &	setTolerance (const RealScalar &tolerance)

Preconditioner &	preconditioner ()

const Preconditioner &	preconditioner () const

Index	maxIterations () const

DGMRES< _MatrixType, _Preconditioner > &	setMaxIterations (Index maxIters)

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

internal::enable_if< Rhs::ColsAtCompileTime!=1 &&DestDerived::ColsAtCompileTime!=1 >::type	_solve_with_guess_impl (const Rhs &b, MatrixBase< DestDerived > &aDest) const

internal::enable_if< Rhs::ColsAtCompileTime==1\|\|DestDerived::ColsAtCompileTime==1 >::type	_solve_with_guess_impl (const Rhs &b, MatrixBase< DestDerived > &dest) const

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

DGMRES< _MatrixType, _Preconditioner > &	derived ()

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

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

	~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

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,.

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

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

template<typename RhsType , typename DestType >
Index	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

Index	m_r

Index	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

bool	m_isInitialized

Protected Attributes inherited from Eigen::SparseSolverBase< Derived >
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, https://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.

Member Typedef Documentation

◆ ComplexVector

template<typename _MatrixType , typename _Preconditioner >

typedef Matrix<std::complex<RealScalar>, Dynamic, 1> Eigen::DGMRES< _MatrixType, _Preconditioner >::ComplexVector

◆ DenseMatrix

template<typename _MatrixType , typename _Preconditioner >

typedef Matrix<Scalar,Dynamic,Dynamic> Eigen::DGMRES< _MatrixType, _Preconditioner >::DenseMatrix

◆ DenseRealMatrix

template<typename _MatrixType , typename _Preconditioner >

typedef Matrix<RealScalar,Dynamic,Dynamic> Eigen::DGMRES< _MatrixType, _Preconditioner >::DenseRealMatrix

◆ DenseRealVector

template<typename _MatrixType , typename _Preconditioner >

typedef Matrix<RealScalar,Dynamic,1> Eigen::DGMRES< _MatrixType, _Preconditioner >::DenseRealVector

◆ DenseVector

template<typename _MatrixType , typename _Preconditioner >

typedef Matrix<Scalar,Dynamic,1> Eigen::DGMRES< _MatrixType, _Preconditioner >::DenseVector

◆ MatrixType

template<typename _MatrixType , typename _Preconditioner >

typedef _MatrixType Eigen::DGMRES< _MatrixType, _Preconditioner >::MatrixType

◆ Preconditioner

template<typename _MatrixType , typename _Preconditioner >

typedef _Preconditioner Eigen::DGMRES< _MatrixType, _Preconditioner >::Preconditioner

◆ RealScalar

template<typename _MatrixType , typename _Preconditioner >

typedef MatrixType::RealScalar Eigen::DGMRES< _MatrixType, _Preconditioner >::RealScalar

◆ Scalar

template<typename _MatrixType , typename _Preconditioner >

typedef MatrixType::Scalar Eigen::DGMRES< _MatrixType, _Preconditioner >::Scalar

◆ StorageIndex

template<typename _MatrixType , typename _Preconditioner >

typedef MatrixType::StorageIndex Eigen::DGMRES< _MatrixType, _Preconditioner >::StorageIndex

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.

◆ ~DGMRES()

template<typename _MatrixType , typename _Preconditioner >

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

inline

Member Function Documentation

◆ _solve_impl()

template<typename _MatrixType , typename _Preconditioner >

template<typename Rhs , typename Dest >

void Eigen::IterativeSolverBase< Derived >::_solve_impl	(	const Rhs &	b,
		Dest &	x
	)		const

inline

◆ _solve_vector_with_guess_impl()

template<typename _MatrixType , typename _Preconditioner >

template<typename Rhs , typename Dest >

void Eigen::DGMRES< _MatrixType, _Preconditioner >::_solve_vector_with_guess_impl	(	const Rhs &	b,
		Dest &	x
	)		const

inline

◆ _solve_with_guess_impl() [1/3]

template<typename _MatrixType , typename _Preconditioner >

template<typename Rhs , typename DestDerived >

internal::enable_if< Rhs::ColsAtCompileTime!=1 &&DestDerived::ColsAtCompileTime!=1 >::type Eigen::IterativeSolverBase< Derived >::_solve_with_guess_impl	(	const Rhs &	b,
		MatrixBase< DestDerived > &	aDest
	)		const

inline

◆ _solve_with_guess_impl() [2/3]

template<typename _MatrixType , typename _Preconditioner >

template<typename Rhs , typename DestDerived >

internal::enable_if< Rhs::ColsAtCompileTime==1\|\|DestDerived::ColsAtCompileTime==1 >::type Eigen::IterativeSolverBase< Derived >::_solve_with_guess_impl	(	const Rhs &	b,
		MatrixBase< DestDerived > &	dest
	)		const

inline

◆ _solve_with_guess_impl() [3/3]

template<typename _MatrixType , typename _Preconditioner >

template<typename Rhs , typename DestDerived >

void Eigen::IterativeSolverBase< Derived >::_solve_with_guess_impl	(	const Rhs &	b,
		SparseMatrixBase< DestDerived > &	aDest
	)		const

inline

◆ deflSize()

template<typename _MatrixType , typename _Preconditioner >

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

inline

Get the size of the deflation subspace size

◆ 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.

◆ dgmresApplyDeflation()

template<typename _MatrixType , typename _Preconditioner >

template<typename RhsType , typename DestType >

Index Eigen::DGMRES< _MatrixType, _Preconditioner >::dgmresApplyDeflation	(	const RhsType &	In,
		DestType &	Out
	)		const

protected

◆ dgmresComputeDeflationData()

template<typename _MatrixType , typename _Preconditioner >

Index Eigen::DGMRES< _MatrixType, _Preconditioner >::dgmresComputeDeflationData	(	const MatrixType &	mat,
		const Preconditioner &	precond,
		const Index &	it,
		StorageIndex &	neig
	)		const

protected

◆ dgmresCycle()

template<typename _MatrixType , typename _Preconditioner >

template<typename Dest >

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

◆ dgmresInitDeflation()

template<typename _MatrixType , typename _Preconditioner >

void Eigen::DGMRES< _MatrixType, _Preconditioner >::dgmresInitDeflation ( Index & rows ) const

protected

◆ restart()

template<typename _MatrixType , typename _Preconditioner >

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

inline

Get the restart value

◆ schurValues() [1/2]

template<typename _MatrixType , typename _Preconditioner >

DGMRES< _MatrixType, _Preconditioner >::ComplexVector Eigen::DGMRES< _MatrixType, _Preconditioner >::schurValues ( const ComplexSchur< DenseMatrix > & schurofH ) const

inlineprotected

◆ schurValues() [2/2]

template<typename _MatrixType , typename _Preconditioner >

DGMRES< _MatrixType, _Preconditioner >::ComplexVector Eigen::DGMRES< _MatrixType, _Preconditioner >::schurValues ( const RealSchur< DenseMatrix > & schurofH ) const

inlineprotected

◆ set_restart()

template<typename _MatrixType , typename _Preconditioner >

void Eigen::DGMRES< _MatrixType, _Preconditioner >::set_restart ( const Index restart )

inline

Set the restart value (default is 30)

◆ setEigenv()

template<typename _MatrixType , typename _Preconditioner >

void Eigen::DGMRES< _MatrixType, _Preconditioner >::setEigenv ( const Index neig )

inline

Set the number of eigenvalues to deflate at each restart

◆ setMaxEigenv()

template<typename _MatrixType , typename _Preconditioner >

void Eigen::DGMRES< _MatrixType, _Preconditioner >::setMaxEigenv ( const Index maxNeig )

inline

Set the maximum size of the deflation subspace

Member Data Documentation

◆ m_force

template<typename _MatrixType , typename _Preconditioner >

bool Eigen::DGMRES< _MatrixType, _Preconditioner >::m_force

mutableprotected

◆ m_H

template<typename _MatrixType , typename _Preconditioner >

DenseMatrix Eigen::DGMRES< _MatrixType, _Preconditioner >::m_H

mutableprotected

◆ m_Hes

template<typename _MatrixType , typename _Preconditioner >

DenseMatrix Eigen::DGMRES< _MatrixType, _Preconditioner >::m_Hes

mutableprotected

◆ m_isDeflAllocated

template<typename _MatrixType , typename _Preconditioner >

bool Eigen::DGMRES< _MatrixType, _Preconditioner >::m_isDeflAllocated

mutableprotected

◆ m_isDeflInitialized

template<typename _MatrixType , typename _Preconditioner >

bool Eigen::DGMRES< _MatrixType, _Preconditioner >::m_isDeflInitialized

mutableprotected

◆ m_lambdaN

template<typename _MatrixType , typename _Preconditioner >

RealScalar Eigen::DGMRES< _MatrixType, _Preconditioner >::m_lambdaN

mutableprotected

◆ m_luT

template<typename _MatrixType , typename _Preconditioner >

PartialPivLU<DenseMatrix> Eigen::DGMRES< _MatrixType, _Preconditioner >::m_luT

mutableprotected

◆ m_maxNeig

template<typename _MatrixType , typename _Preconditioner >

Index Eigen::DGMRES< _MatrixType, _Preconditioner >::m_maxNeig

mutableprotected

◆ m_MU

template<typename _MatrixType , typename _Preconditioner >

DenseMatrix Eigen::DGMRES< _MatrixType, _Preconditioner >::m_MU

mutableprotected

◆ m_neig

template<typename _MatrixType , typename _Preconditioner >

StorageIndex Eigen::DGMRES< _MatrixType, _Preconditioner >::m_neig

mutableprotected

◆ m_r

template<typename _MatrixType , typename _Preconditioner >

Index Eigen::DGMRES< _MatrixType, _Preconditioner >::m_r

mutableprotected

◆ m_restart

template<typename _MatrixType , typename _Preconditioner >

Index Eigen::DGMRES< _MatrixType, _Preconditioner >::m_restart

mutableprotected

◆ m_smv

template<typename _MatrixType , typename _Preconditioner >

RealScalar Eigen::DGMRES< _MatrixType, _Preconditioner >::m_smv

mutableprotected

◆ m_T

template<typename _MatrixType , typename _Preconditioner >

DenseMatrix Eigen::DGMRES< _MatrixType, _Preconditioner >::m_T

mutableprotected

◆ m_U

template<typename _MatrixType , typename _Preconditioner >

DenseMatrix Eigen::DGMRES< _MatrixType, _Preconditioner >::m_U

mutableprotected

◆ m_V

template<typename _MatrixType , typename _Preconditioner >

DenseMatrix Eigen::DGMRES< _MatrixType, _Preconditioner >::m_V

mutableprotected

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

core/util/algorithms/eigen-3.4.0/unsupported/Eigen/src/IterativeSolvers/DGMRES.h

Public Types

Public Member Functions

Protected Member Functions

Protected Attributes

Additional Inherited Members

Detailed Description

Member Typedef Documentation

◆ ComplexVector

◆ DenseMatrix

◆ DenseRealMatrix

◆ DenseRealVector

◆ DenseVector

◆ MatrixType

◆ Preconditioner

◆ RealScalar

◆ Scalar

◆ StorageIndex

Constructor & Destructor Documentation

◆ DGMRES() [1/2]

◆ DGMRES() [2/2]

◆ ~DGMRES()

Member Function Documentation

◆ _solve_impl()

◆ _solve_vector_with_guess_impl()

◆ _solve_with_guess_impl() [1/3]

◆ _solve_with_guess_impl() [2/3]

◆ _solve_with_guess_impl() [3/3]

◆ deflSize()

◆ dgmres()

◆ dgmresApplyDeflation()

◆ dgmresComputeDeflationData()

◆ dgmresCycle()

◆ dgmresInitDeflation()

◆ restart()

◆ schurValues() [1/2]

◆ schurValues() [2/2]

◆ set_restart()

◆ setEigenv()

◆ setMaxEigenv()

Member Data Documentation

◆ m_force

◆ m_H

◆ m_Hes

◆ m_isDeflAllocated

◆ m_isDeflInitialized

◆ m_lambdaN

◆ m_luT

◆ m_maxNeig

◆ m_MU

◆ m_neig

◆ m_r

◆ m_restart

◆ m_smv

◆ m_T

◆ m_U

◆ m_V