Public Types | Public Member Functions | Protected Member Functions | Protected Attributes | Private Types | List of all members
Eigen::DGMRES< _MatrixType, _Preconditioner > Class Template Reference

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 >:
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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::Index Index
 
typedef _MatrixType MatrixType
 
typedef _Preconditioner Preconditioner
 
typedef MatrixType::RealScalar RealScalar
 
typedef MatrixType::Scalar Scalar
 
- Public Types inherited from Eigen::IterativeSolverBase< DGMRES< _MatrixType, _Preconditioner > >
typedef MatrixType::Index Index
 
typedef internal::traits< DGMRES< _MatrixType, _Preconditioner > >::MatrixType MatrixType
 
typedef internal::traits< DGMRES< _MatrixType, _Preconditioner > >::Preconditioner Preconditioner
 
typedef MatrixType::RealScalar RealScalar
 
typedef MatrixType::Scalar Scalar
 

Public Member Functions

template<typename Rhs , typename Dest >
void _solve (const Rhs &b, Dest &x) const
 
template<typename Rhs , typename Dest >
void _solveWithGuess (const Rhs &b, Dest &x) const
 
int deflSize ()
 
 DGMRES ()
 
 DGMRES (const MatrixType &A)
 
int restart ()
 
void set_restart (const int restart)
 
void setEigenv (const int neig)
 
void setMaxEigenv (const int maxNeig)
 
template<typename Rhs , typename Guess >
const internal::solve_retval_with_guess< DGMRES, Rhs, Guess > solveWithGuess (const MatrixBase< Rhs > &b, const Guess &x0) const
 
 ~DGMRES ()
 
- Public Member Functions inherited from Eigen::IterativeSolverBase< DGMRES< _MatrixType, _Preconditioner > >
void _solve_sparse (const Rhs &b, SparseMatrix< DestScalar, DestOptions, DestIndex > &dest) const
 
DGMRES< _MatrixType, _Preconditioner > & analyzePattern (const MatrixType &A)
 
Index cols () const
 
DGMRES< _MatrixType, _Preconditioner > & compute (const MatrixType &A)
 
DGMRES< _MatrixType, _Preconditioner > & derived ()
 
const DGMRES< _MatrixType, _Preconditioner > & derived () const
 
RealScalar error () const
 
DGMRES< _MatrixType, _Preconditioner > & factorize (const MatrixType &A)
 
ComputationInfo info () const
 
int iterations () const
 
 IterativeSolverBase ()
 
 IterativeSolverBase (const MatrixType &A)
 
int maxIterations () const
 
Preconditioner & preconditioner ()
 
const Preconditioner & preconditioner () const
 
Index rows () const
 
DGMRES< _MatrixType, _Preconditioner > & setMaxIterations (int maxIters)
 
DGMRES< _MatrixType, _Preconditioner > & setTolerance (const RealScalar &tolerance)
 
const internal::solve_retval< DGMRES< _MatrixType, _Preconditioner >, Rhs > solve (const MatrixBase< Rhs > &b) const
 
const internal::sparse_solve_retval< IterativeSolverBase, Rhs > solve (const SparseMatrixBase< Rhs > &b) const
 
RealScalar tolerance () const
 
 ~IterativeSolverBase ()
 

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 >
int dgmresApplyDeflation (const RhsType &In, DestType &Out) const
 
int dgmresComputeDeflationData (const MatrixType &mat, const Preconditioner &precond, const Index &it, Index &neig) const
 
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...
 
void dgmresInitDeflation (Index &rows) const
 
ComplexVector schurValues (const ComplexSchur< DenseMatrix > &schurofH) const
 
ComplexVector schurValues (const RealSchur< DenseMatrix > &schurofH) const
 
- Protected Member Functions inherited from Eigen::IterativeSolverBase< DGMRES< _MatrixType, _Preconditioner > >
void init ()
 

Protected Attributes

bool m_force
 
DenseMatrix m_H
 
DenseMatrix m_Hes
 
bool m_isDeflAllocated
 
bool m_isDeflInitialized
 
RealScalar m_lambdaN
 
PartialPivLU< DenseMatrix > m_luT
 
int m_maxNeig
 
DenseMatrix m_MU
 
int m_neig
 
int m_r
 
Index m_restart
 
RealScalar m_smv
 
DenseMatrix m_T
 
DenseMatrix m_U
 
DenseMatrix m_V
 
- Protected Attributes inherited from Eigen::IterativeSolverBase< DGMRES< _MatrixType, _Preconditioner > >
bool m_analysisIsOk
 
RealScalar m_error
 
bool m_factorizationIsOk
 
ComputationInfo m_info
 
bool m_isInitialized
 
int m_iterations
 
int m_maxIterations
 
Preconditioner m_preconditioner
 
RealScalar m_tolerance
 
const MatrixType * mp_matrix
 

Private Types

typedef IterativeSolverBase< DGMRES > Base
 

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
_MatrixTypethe type of the sparse matrix A, can be a dense or a sparse matrix.
_Preconditionerthe 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);

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.

Definition at line 19 of file DGMRES.h.

Member Typedef Documentation

template<typename _MatrixType , typename _Preconditioner >
typedef IterativeSolverBase<DGMRES> Eigen::DGMRES< _MatrixType, _Preconditioner >::Base
private

Definition at line 103 of file DGMRES.h.

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

Definition at line 120 of file DGMRES.h.

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

Definition at line 116 of file DGMRES.h.

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

Definition at line 117 of file DGMRES.h.

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

Definition at line 119 of file DGMRES.h.

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

Definition at line 118 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
typedef MatrixType::Index Eigen::DGMRES< _MatrixType, _Preconditioner >::Index

Definition at line 113 of file DGMRES.h.

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

Definition at line 111 of file DGMRES.h.

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

Definition at line 115 of file DGMRES.h.

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

Definition at line 114 of file DGMRES.h.

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

Definition at line 112 of file DGMRES.h.

Constructor & Destructor Documentation

template<typename _MatrixType , typename _Preconditioner >
Eigen::DGMRES< _MatrixType, _Preconditioner >::DGMRES ( )
inline

Default constructor.

Definition at line 124 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
Eigen::DGMRES< _MatrixType, _Preconditioner >::DGMRES ( const MatrixType &  A)
inline

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 136 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
Eigen::DGMRES< _MatrixType, _Preconditioner >::~DGMRES ( )
inline

Definition at line 139 of file DGMRES.h.

Member Function Documentation

template<typename _MatrixType , typename _Preconditioner >
template<typename Rhs , typename Dest >
void Eigen::DGMRES< _MatrixType, _Preconditioner >::_solve ( const Rhs &  b,
Dest &  x 
) const
inline

Definition at line 178 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
template<typename Rhs , typename Dest >
void Eigen::DGMRES< _MatrixType, _Preconditioner >::_solveWithGuess ( const Rhs &  b,
Dest &  x 
) const
inline

Definition at line 159 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
int Eigen::DGMRES< _MatrixType, _Preconditioner >::deflSize ( )
inline

Get the size of the deflation subspace size

Definition at line 205 of file DGMRES.h.

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 256 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
template<typename RhsType , typename DestType >
int Eigen::DGMRES< _MatrixType, _Preconditioner >::dgmresApplyDeflation ( const RhsType &  In,
DestType &  Out 
) const
protected

Definition at line 518 of file DGMRES.h.

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

Definition at line 434 of file DGMRES.h.

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
matThe coefficient matrix
precondThe preconditioner
xthe new approximated solution
r0The initial residual vector
betaThe norm of the residual computed so far
normRhsThe norm of the right hand side vector
nbItsThe number of iterations

Definition at line 300 of file DGMRES.h.

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

Definition at line 392 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
int Eigen::DGMRES< _MatrixType, _Preconditioner >::restart ( )
inline

Get the restart value

Definition at line 186 of file DGMRES.h.

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

Definition at line 402 of file DGMRES.h.

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

Definition at line 408 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
void Eigen::DGMRES< _MatrixType, _Preconditioner >::set_restart ( const int  restart)
inline

Set the restart value (default is 30)

Definition at line 191 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
void Eigen::DGMRES< _MatrixType, _Preconditioner >::setEigenv ( const int  neig)
inline

Set the number of eigenvalues to deflate at each restart

Definition at line 196 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
void Eigen::DGMRES< _MatrixType, _Preconditioner >::setMaxEigenv ( const int  maxNeig)
inline

Set the maximum size of the deflation subspace

Definition at line 210 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
template<typename Rhs , typename Guess >
const internal::solve_retval_with_guess<DGMRES, Rhs, Guess> Eigen::DGMRES< _MatrixType, _Preconditioner >::solveWithGuess ( const MatrixBase< Rhs > &  b,
const Guess &  x0 
) const
inline
Returns
the solution x of $ A x = b $ using the current decomposition of A x0 as an initial solution.
See also
compute()

Definition at line 148 of file DGMRES.h.

Member Data Documentation

template<typename _MatrixType , typename _Preconditioner >
bool Eigen::DGMRES< _MatrixType, _Preconditioner >::m_force
mutableprotected

Definition at line 245 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
DenseMatrix Eigen::DGMRES< _MatrixType, _Preconditioner >::m_H
mutableprotected

Definition at line 229 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
DenseMatrix Eigen::DGMRES< _MatrixType, _Preconditioner >::m_Hes
mutableprotected

Definition at line 230 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
bool Eigen::DGMRES< _MatrixType, _Preconditioner >::m_isDeflAllocated
mutableprotected

Definition at line 240 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
bool Eigen::DGMRES< _MatrixType, _Preconditioner >::m_isDeflInitialized
mutableprotected

Definition at line 241 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
RealScalar Eigen::DGMRES< _MatrixType, _Preconditioner >::m_lambdaN
mutableprotected

Definition at line 239 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
PartialPivLU<DenseMatrix> Eigen::DGMRES< _MatrixType, _Preconditioner >::m_luT
mutableprotected

Definition at line 235 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
int Eigen::DGMRES< _MatrixType, _Preconditioner >::m_maxNeig
mutableprotected

Definition at line 238 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
DenseMatrix Eigen::DGMRES< _MatrixType, _Preconditioner >::m_MU
mutableprotected

Definition at line 233 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
int Eigen::DGMRES< _MatrixType, _Preconditioner >::m_neig
mutableprotected

Definition at line 236 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
int Eigen::DGMRES< _MatrixType, _Preconditioner >::m_r
mutableprotected

Definition at line 237 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
Index Eigen::DGMRES< _MatrixType, _Preconditioner >::m_restart
mutableprotected

Definition at line 231 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
RealScalar Eigen::DGMRES< _MatrixType, _Preconditioner >::m_smv
mutableprotected

Definition at line 244 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
DenseMatrix Eigen::DGMRES< _MatrixType, _Preconditioner >::m_T
mutableprotected

Definition at line 234 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
DenseMatrix Eigen::DGMRES< _MatrixType, _Preconditioner >::m_U
mutableprotected

Definition at line 232 of file DGMRES.h.

template<typename _MatrixType , typename _Preconditioner >
DenseMatrix Eigen::DGMRES< _MatrixType, _Preconditioner >::m_V
mutableprotected

Definition at line 228 of file DGMRES.h.

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

acado
Author(s): Milan Vukov, Rien Quirynen
autogenerated on Mon Jun 10 2019 12:35:33