#include <deal.II/lac/solver_gmres.h>

Inheritance diagram for SolverGMRES< VectorType >:

Classes
struct	AdditionalData

Public Member Functions
	SolverGMRES (SolverControl &cn, VectorMemory< VectorType > &mem, const AdditionalData &data=AdditionalData())

	SolverGMRES (SolverControl &cn, const AdditionalData &data=AdditionalData())

template<typename MatrixType , typename PreconditionerType >
void	solve (const MatrixType &A, VectorType &x, const VectorType &b, const PreconditionerType &preconditioner)

boost::signals2::connection	connect_condition_number_slot (const std::function< void(double)> &slot, const bool every_iteration=false)

boost::signals2::connection	connect_eigenvalues_slot (const std::function< void(const std::vector< std::complex< double > > &)> &slot, const bool every_iteration=false)

boost::signals2::connection	connect_hessenberg_slot (const std::function< void(const FullMatrix< double > &)> &slot, const bool every_iteration=true)

boost::signals2::connection	connect_krylov_space_slot (const std::function< void(const internal::SolverGMRESImplementation::TmpVectors< VectorType > &)> &slot)

boost::signals2::connection	connect_re_orthogonalization_slot (const std::function< void(int)> &slot)

Public Member Functions inherited from Solver< VectorType >
	Solver (SolverControl &solver_control, VectorMemory< VectorType > &vector_memory)

	Solver (SolverControl &solver_control)

boost::signals2::connection	connect (const std::function< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType &current_iterate)> &slot)

Public Member Functions inherited from Subscriptor
	Subscriptor ()

	Subscriptor (const Subscriptor &)

	Subscriptor (Subscriptor &&) noexcept

virtual	~Subscriptor ()

Subscriptor &	operator= (const Subscriptor &)

Subscriptor &	operator= (Subscriptor &&) noexcept

void	subscribe (const char *identifier=nullptr) const

void	unsubscribe (const char *identifier=nullptr) const

unsigned int	n_subscriptions () const

void	list_subscribers () const

template<class Archive >
void	serialize (Archive &ar, const unsigned int version)

Static Public Member Functions
static ::ExceptionBase &	ExcTooFewTmpVectors (int arg1)

Static Public Member Functions inherited from Subscriptor
static ::ExceptionBase &	ExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBase &	ExcNoSubscriber (std::string arg1, std::string arg2)

Protected Member Functions
virtual double	criterion ()

void	givens_rotation (Vector< double > &h, Vector< double > &b, Vector< double > &ci, Vector< double > &si, int col) const

Static Protected Member Functions
static double	modified_gram_schmidt (const internal::SolverGMRESImplementation::TmpVectors< VectorType > &orthogonal_vectors, const unsigned int dim, const unsigned int accumulated_iterations, VectorType &vv, Vector< double > &h, bool &re_orthogonalize, const boost::signals2::signal< void(int)> &re_orthogonalize_signal=boost::signals2::signal< void(int)>())

static void	compute_eigs_and_cond (const FullMatrix< double > &H_orig, const unsigned int dim, const boost::signals2::signal< void(const std::vector< std::complex< double > > &)> &eigenvalues_signal, const boost::signals2::signal< void(const FullMatrix< double > &)> &hessenberg_signal, const boost::signals2::signal< void(double)> &cond_signal)

Protected Attributes
AdditionalData	additional_data

boost::signals2::signal< void(double)>	condition_number_signal

boost::signals2::signal< void(double)>	all_condition_numbers_signal

boost::signals2::signal< void(const std::vector< std::complex< double > > &)>	eigenvalues_signal

boost::signals2::signal< void(const std::vector< std::complex< double > > &)>	all_eigenvalues_signal

boost::signals2::signal< void(const FullMatrix< double > &)>	hessenberg_signal

boost::signals2::signal< void(const FullMatrix< double > &)>	all_hessenberg_signal

boost::signals2::signal< void(const internal::SolverGMRESImplementation::TmpVectors< VectorType > &)>	krylov_space_signal

boost::signals2::signal< void(int)>	re_orthogonalize_signal

FullMatrix< double >	H

FullMatrix< double >	H1

Protected Attributes inherited from Solver< VectorType >
GrowingVectorMemory< VectorType >	static_vector_memory

VectorMemory< VectorType > &	memory

boost::signals2::signal< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType &current_iterate), StateCombiner >	iteration_status

Private Member Functions
	SolverGMRES (const SolverGMRES< VectorType > &)

Additional Inherited Members
Public Types inherited from Solver< VectorType >
typedef VectorType	vector_type

Detailed Description

template<class VectorType = Vector<double>>
class SolverGMRES< VectorType >

Implementation of the Restarted Preconditioned Direct Generalized Minimal Residual Method. The stopping criterion is the norm of the residual.

The AdditionalData structure contains the number of temporary vectors used. The size of the Arnoldi basis is this number minus three. Additionally, it allows you to choose between right or left preconditioning. The default is left preconditioning. Finally it includes a flag indicating whether or not the default residual is used as stopping criterion.

Left versus right preconditioning

AdditionalData allows you to choose between left and right preconditioning. As expected, this switches between solving for the systems P^-1A and AP^-1, respectively.

A second consequence is the type of residual which is used to measure convergence. With left preconditioning, this is the preconditioned residual, while with right preconditioning, it is the residual of the unpreconditioned system.

Optionally, this behavior can be overridden by using the flag AdditionalData::use_default_residual. A true value refers to the behavior described in the previous paragraph, while false reverts it. Be aware though that additional residuals have to be computed in this case, impeding the overall performance of the solver.

The size of the Arnoldi basis

The maximal basis size is controlled by AdditionalData::max_n_tmp_vectors, and it is this number minus 2. If the number of iteration steps exceeds this number, all basis vectors are discarded and the iteration starts anew from the approximation obtained so far.

Note that the minimizing property of GMRes only pertains to the Krylov space spanned by the Arnoldi basis. Therefore, restarted GMRes is not minimizing anymore. The choice of the basis length is a trade- off between memory consumption and convergence speed, since a longer basis means minimization over a larger space.

For the requirements on matrices and vectors in order to work with this class, see the documentation of the Solver base class.

Observing the progress of linear solver iterations

The solve() function of this class uses the mechanism described in the Solver base class to determine convergence. This mechanism can also be used to observe the progress of the iteration.

Eigenvalue and condition number estimates

This class can estimate eigenvalues and condition number during the solution process. This is done by creating the Hessenberg matrix during the inner iterations. The eigenvalues are estimated as the eigenvalues of the Hessenberg matrix and the condition number is estimated as the ratio of the largest and smallest singular value of the Hessenberg matrix. The estimates can be obtained by connecting a function as a slot using connect_condition_number_slot and connect_eigenvalues_slot. These slots will then be called from the solver with the estimates as argument.

Author: Wolfgang Bangerth, Guido Kanschat, Ralf Hartmann.

Definition at line 177 of file solver_gmres.h.