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>

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

typedef Matrix< Scalar, Dynamic, Dynamic >	DenseMatrix

typedef Matrix< RealScalar, Dynamic, Dynamic >	DenseRealMatrix

typedef Matrix< RealScalar, Dynamic, 1 >	DenseRealVector

typedef Matrix< Scalar, Dynamic, 1 >	DenseVector

typedef _MatrixType	MatrixType

typedef _Preconditioner	Preconditioner

typedef MatrixType::RealScalar	RealScalar

typedef MatrixType::Scalar	Scalar

typedef MatrixType::StorageIndex	StorageIndex

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

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

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

Index	deflSize ()

	DGMRES ()

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

Index	restart ()

void	set_restart (const Index restart)

void	setEigenv (const Index neig)

void	setMaxEigenv (const Index maxNeig)

	~DGMRES ()

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 RhsType , typename DestType >
Index	dgmresApplyDeflation (const RhsType &In, DestType &Out) const

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

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. More...

void	dgmresInitDeflation (Index &rows) const

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

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

Protected Attributes
bool	m_force

DenseMatrix	m_H

DenseMatrix	m_Hes

bool	m_isDeflAllocated

bool	m_isDeflInitialized

RealScalar	m_lambdaN

PartialPivLU< DenseMatrix >	m_luT

Index	m_maxNeig

DenseMatrix	m_MU

StorageIndex	m_neig

Index	m_r

Index	m_restart

RealScalar	m_smv

DenseMatrix	m_T

DenseMatrix	m_U

DenseMatrix	m_V

Private Types
typedef IterativeSolverBase< DGMRES >	Base

Private Member Functions
const ActualMatrixType &	matrix () const

Private Attributes
RealScalar	m_error

ComputationInfo	m_info

bool	m_isInitialized

Index	m_iterations

RealScalar	m_tolerance

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.

Definition at line 19 of file DGMRES.h.

Member Typedef Documentation

◆ Base

template<typename _MatrixType , typename _Preconditioner >

typedef IterativeSolverBase<DGMRES> Eigen::DGMRES< _MatrixType, _Preconditioner >::Base

private

Definition at line 103 of file DGMRES.h.

◆ ComplexVector

template<typename _MatrixType , typename _Preconditioner >

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

Definition at line 122 of file DGMRES.h.

◆ DenseMatrix

template<typename _MatrixType , typename _Preconditioner >

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

Definition at line 118 of file DGMRES.h.

◆ DenseRealMatrix

template<typename _MatrixType , typename _Preconditioner >

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

Definition at line 119 of file DGMRES.h.

◆ DenseRealVector

template<typename _MatrixType , typename _Preconditioner >

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

Definition at line 121 of file DGMRES.h.

◆ DenseVector

template<typename _MatrixType , typename _Preconditioner >

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

Definition at line 120 of file DGMRES.h.

◆ MatrixType

template<typename _MatrixType , typename _Preconditioner >

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

Definition at line 113 of file DGMRES.h.

◆ Preconditioner

template<typename _MatrixType , typename _Preconditioner >

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

Definition at line 117 of file DGMRES.h.

◆ RealScalar

template<typename _MatrixType , typename _Preconditioner >

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

Definition at line 116 of file DGMRES.h.

◆ Scalar

template<typename _MatrixType , typename _Preconditioner >

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

Definition at line 114 of file DGMRES.h.

◆ StorageIndex

template<typename _MatrixType , typename _Preconditioner >

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

Definition at line 115 of file DGMRES.h.

Constructor & Destructor Documentation

◆ DGMRES() [1/2]

template<typename _MatrixType , typename _Preconditioner >

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

inline

Default constructor.

Definition at line 126 of file DGMRES.h.

◆ 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.

Definition at line 139 of file DGMRES.h.

◆ ~DGMRES()

template<typename _MatrixType , typename _Preconditioner >

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

inline

Definition at line 141 of file DGMRES.h.

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

Definition at line 400 of file IterativeSolverBase.h.

◆ _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

Definition at line 145 of file DGMRES.h.

◆ _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	(	typename Rhs	,
		typename DestDerived
	)

inline

Definition at line 368 of file IterativeSolverBase.h.

◆ _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	(	typename Rhs	,
		typename DestDerived
	)

inline

Definition at line 393 of file IterativeSolverBase.h.

◆ _solve_with_guess_impl() [3/3]

template<typename _MatrixType , typename _Preconditioner >

template<typename Rhs , typename DestDerived >

void Eigen::IterativeSolverBase< Derived >::_solve_with_guess_impl	(	typename Rhs	,
		typename DestDerived
	)

inline

Definition at line 334 of file IterativeSolverBase.h.

◆ deflSize()

template<typename _MatrixType , typename _Preconditioner >

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

inline

Get the size of the deflation subspace size

Definition at line 177 of file DGMRES.h.

◆ 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.

Definition at line 228 of file DGMRES.h.

◆ dgmresApplyDeflation()

template<typename _MatrixType , typename _Preconditioner >

template<typename RhsType , typename DestType >

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

protected

Definition at line 503 of file DGMRES.h.

◆ 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

Definition at line 419 of file DGMRES.h.

◆ 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

Definition at line 286 of file DGMRES.h.

◆ dgmresInitDeflation()

template<typename _MatrixType , typename _Preconditioner >

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

protected

Definition at line 378 of file DGMRES.h.

◆ matrix()

template<typename _MatrixType , typename _Preconditioner >

const ActualMatrixType& Eigen::IterativeSolverBase< Derived >::matrix

inlineprivate

Definition at line 419 of file IterativeSolverBase.h.

◆ restart()

template<typename _MatrixType , typename _Preconditioner >

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

inline

Get the restart value

Definition at line 158 of file DGMRES.h.

◆ schurValues() [1/2]

template<typename _MatrixType , typename _Preconditioner >

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

inlineprotected

Definition at line 388 of file DGMRES.h.

◆ schurValues() [2/2]

template<typename _MatrixType , typename _Preconditioner >

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

inlineprotected

Definition at line 394 of file DGMRES.h.

◆ 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)

Definition at line 163 of file DGMRES.h.

◆ 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

Definition at line 168 of file DGMRES.h.

◆ setMaxEigenv()

template<typename _MatrixType , typename _Preconditioner >

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

inline

Set the maximum size of the deflation subspace

Definition at line 182 of file DGMRES.h.

Member Data Documentation

◆ m_error

template<typename _MatrixType , typename _Preconditioner >

RealScalar Eigen::IterativeSolverBase< Derived >::m_error

mutableprivate

Definition at line 436 of file IterativeSolverBase.h.

◆ m_force

template<typename _MatrixType , typename _Preconditioner >

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

mutableprotected

Definition at line 217 of file DGMRES.h.

◆ m_H

template<typename _MatrixType , typename _Preconditioner >

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

mutableprotected

Definition at line 201 of file DGMRES.h.

◆ m_Hes

template<typename _MatrixType , typename _Preconditioner >

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

mutableprotected

Definition at line 202 of file DGMRES.h.

◆ m_info

template<typename _MatrixType , typename _Preconditioner >

ComputationInfo Eigen::IterativeSolverBase< Derived >::m_info

mutableprivate

Definition at line 438 of file IterativeSolverBase.h.

◆ m_isDeflAllocated

template<typename _MatrixType , typename _Preconditioner >

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

mutableprotected

Definition at line 212 of file DGMRES.h.

◆ m_isDeflInitialized

template<typename _MatrixType , typename _Preconditioner >

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

mutableprotected

Definition at line 213 of file DGMRES.h.

◆ m_isInitialized

template<typename _MatrixType , typename _Preconditioner >

bool Eigen::SparseSolverBase< Derived >::m_isInitialized

mutableprivate

Definition at line 119 of file SparseSolverBase.h.

◆ m_iterations

template<typename _MatrixType , typename _Preconditioner >

Index Eigen::IterativeSolverBase< Derived >::m_iterations

mutableprivate

Definition at line 437 of file IterativeSolverBase.h.

◆ m_lambdaN

template<typename _MatrixType , typename _Preconditioner >

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

mutableprotected

Definition at line 211 of file DGMRES.h.

◆ m_luT

template<typename _MatrixType , typename _Preconditioner >

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

mutableprotected

Definition at line 207 of file DGMRES.h.

◆ m_maxNeig

template<typename _MatrixType , typename _Preconditioner >

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

mutableprotected

Definition at line 210 of file DGMRES.h.

◆ m_MU

template<typename _MatrixType , typename _Preconditioner >

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

mutableprotected

Definition at line 205 of file DGMRES.h.

◆ m_neig

template<typename _MatrixType , typename _Preconditioner >

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

mutableprotected

Definition at line 208 of file DGMRES.h.

◆ m_r

template<typename _MatrixType , typename _Preconditioner >

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

mutableprotected

Definition at line 209 of file DGMRES.h.

◆ m_restart

template<typename _MatrixType , typename _Preconditioner >

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

mutableprotected

Definition at line 203 of file DGMRES.h.

◆ m_smv

template<typename _MatrixType , typename _Preconditioner >

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

mutableprotected

Definition at line 216 of file DGMRES.h.

◆ m_T

template<typename _MatrixType , typename _Preconditioner >

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

mutableprotected

Definition at line 206 of file DGMRES.h.

◆ m_tolerance

template<typename _MatrixType , typename _Preconditioner >

RealScalar Eigen::IterativeSolverBase< Derived >::m_tolerance

private

Definition at line 434 of file IterativeSolverBase.h.

◆ m_U

template<typename _MatrixType , typename _Preconditioner >

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

mutableprotected

Definition at line 204 of file DGMRES.h.

◆ m_V

template<typename _MatrixType , typename _Preconditioner >

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

mutableprotected

Definition at line 200 of file DGMRES.h.

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

DGMRES.h

Public Types

Public Member Functions

Protected Member Functions

Protected Attributes

Private Types

Private Member Functions

Private Attributes

Detailed Description

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

Member Typedef Documentation

◆ Base

◆ 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()

◆ matrix()

◆ restart()

◆ schurValues() [1/2]

◆ schurValues() [2/2]

◆ set_restart()

◆ setEigenv()

◆ setMaxEigenv()

Member Data Documentation

◆ m_error

◆ m_force

◆ m_H

◆ m_Hes

◆ m_info

◆ m_isDeflAllocated

◆ m_isDeflInitialized

◆ m_isInitialized

◆ m_iterations

◆ m_lambdaN

◆ m_luT

◆ m_maxNeig

◆ m_MU

◆ m_neig

◆ m_r

◆ m_restart

◆ m_smv

◆ m_T

◆ m_tolerance

◆ m_U

◆ m_V

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