Eigen::DGMRES< MatrixType_, Preconditioner_ > Class Template Reference

#include <Eigen/src/IterativeLinearSolvers/DGMRES.h>

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.

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

Public Member Functions
Index	deflSize () const
	DGMRES ()
template<typename MatrixDerived>
	DGMRES (const EigenBase< MatrixDerived > &A)
Index	restart () const
void	set_restart (const Index restart)
void	setEigenv (const Index neig)
void	setMaxEigenv (const Index maxNeig)
Public Member Functions inherited from Eigen::IterativeSolverBase< DGMRES< MatrixType_, Preconditioner_ > >
DGMRES< MatrixType_, Preconditioner_ > &	analyzePattern (const EigenBase< MatrixDerived > &A)
DGMRES< MatrixType_, Preconditioner_ > &	compute (const EigenBase< MatrixDerived > &A)
RealScalar	error () const
DGMRES< MatrixType_, Preconditioner_ > &	factorize (const EigenBase< MatrixDerived > &A)
ComputationInfo	info () const
Index	iterations () const
	IterativeSolverBase ()
	IterativeSolverBase (const EigenBase< MatrixDerived > &A)
Index	maxIterations () const
Preconditioner &	preconditioner ()
const Preconditioner &	preconditioner () const
DGMRES< MatrixType_, Preconditioner_ > &	setMaxIterations (Index maxIters)
DGMRES< MatrixType_, Preconditioner_ > &	setTolerance (const RealScalar &tolerance)
const SolveWithGuess< DGMRES< MatrixType_, Preconditioner_ >, Rhs, Guess >	solveWithGuess (const MatrixBase< Rhs > &b, const Guess &x0) const
void	solveWithGuessInPlace (const Rhs &b, Dest &x) const
RealScalar	tolerance () const
Public Member Functions inherited from Eigen::SparseSolverBase< DGMRES< MatrixType_, Preconditioner_ > >
Solve< DGMRES< MatrixType_, Preconditioner_ >, Rhs >	solve (const MatrixBase< Rhs > &b) const
Solve< DGMRES< MatrixType_, Preconditioner_ >, Rhs >	solve (const SparseMatrixBase< Rhs > &b) const
	SparseSolverBase ()=default

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.

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

deflSize()

template<typename MatrixType_, typename Preconditioner_>

Index Eigen::DGMRES< MatrixType_, Preconditioner_ >::deflSize ( ) const

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.

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

restart()

template<typename MatrixType_, typename Preconditioner_>

Index Eigen::DGMRES< MatrixType_, Preconditioner_ >::restart ( ) const

inline

Get the restart value

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

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

• /builds/libeigen/eigen/Eigen/src/IterativeLinearSolvers/DGMRES.h