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

Inheritance diagram for SolverCG< VectorType >:

Classes
struct	AdditionalData

Public Types
typedef types::global_dof_index	size_type

Public Types inherited from Solver< VectorType >
typedef VectorType	vector_type

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

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

virtual	~SolverCG ()=default

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

boost::signals2::connection	connect_coefficients_slot (const std::function< void(double, double)> &slot)

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< double > &)> &slot, const bool every_iteration=false)

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)

Protected Member Functions
virtual void	print_vectors (const unsigned int step, const VectorType &x, const VectorType &r, const VectorType &d) const

Static Protected Member Functions
static void	compute_eigs_and_cond (const std::vector< double > &diagonal, const std::vector< double > &offdiagonal, const boost::signals2::signal< void(const std::vector< double > &)> &eigenvalues_signal, const boost::signals2::signal< void(double)> &cond_signal)

Protected Attributes
AdditionalData	additional_data

boost::signals2::signal< void(double, double)>	coefficients_signal

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

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

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

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

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

Additional Inherited Members
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)

Detailed Description

template<typename VectorType = Vector<double>>
class SolverCG< VectorType >

This class implements the preconditioned Conjugate Gradients (CG) method that can be used to solve linear systems with a symmetric positive definite matrix. This class is used first in step-3 and step-4, but is used in many other tutorial programs as well. Like all other solver classes, it can work on any kind of vector and matrix as long as they satisfy certain requirements (for the requirements on matrices and vectors in order to work with this class, see the documentation of the Solver base class). The type of the solution vector must be passed as template argument, and defaults to Vector<double>.

Note: This version of CG is taken from D. Braess's book "Finite Elements". It requires a symmetric preconditioner (i.e., for example, SOR is not a possible choice).

Eigenvalue computation

The cg-method performs an orthogonal projection of the original preconditioned linear system to another system of smaller dimension. Furthermore, the projected matrix T is tri-diagonal. Since the projection is orthogonal, the eigenvalues of T approximate those of the original preconditioned matrix PA. In fact, after n steps, where n is the dimension of the original system, the eigenvalues of both matrices are equal. But, even for small numbers of iteration steps, the condition number of T is a good estimate for the one of PA.

After m steps the matrix T_m can be written in terms of the coefficients alpha and beta as the tri-diagonal matrix with diagonal elements 1/alpha_0, 1/alpha_1 + beta_0/alpha_0, ..., 1/alpha_{m-1+beta_{m-2}/alpha_{m-2}} and off-diagonal elements sqrt(beta_0)/alpha_0, ..., sqrt(beta_{m-2)/alpha_{m-2}}. The eigenvalues of this matrix can be computed by postprocessing.

See also: Y. Saad: "Iterative methods for Sparse Linear Systems", section 6.7.3 for details.

The coefficients, eigenvalues and condition number (computed as the ratio of the largest over smallest eigenvalue) can be obtained by connecting a function as a slot to the solver using one of the functions connect_coefficients_slot, connect_eigenvalues_slot and connect_condition_number_slot. These slots will then be called from the solver with the estimates as argument.

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.

Author: W. Bangerth, G. Kanschat, R. Becker and F.-T. Suttmeier

Definition at line 93 of file solver_cg.h.