Reference documentation for deal.II version 9.1.1
\(\newcommand{\dealcoloneq}{\mathrel{\vcenter{:}}=}\)
Classes | Public Types | Public Member Functions | Protected Member Functions | Static Protected Member Functions | Protected Attributes | List of all members
SolverCG< VectorType > Class Template Reference

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

Inheritance diagram for SolverCG< VectorType >:
[legend]

Classes

struct  AdditionalData
 

Public Types

using size_type = types::global_dof_index
 
- Public Types inherited from SolverBase< VectorType >
using vector_type = VectorType
 

Public Member Functions

 SolverCG (SolverControl &cn, VectorMemory< VectorType > &mem, const AdditionalData &data=AdditionalData())
 
 SolverCG (SolverControl &cn, const AdditionalData &data=AdditionalData())
 
virtual ~SolverCG () override=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(typename VectorType::value_type, typename VectorType::value_type)> &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 SolverBase< VectorType >
 SolverBase (SolverControl &solver_control, VectorMemory< VectorType > &vector_memory)
 
 SolverBase (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 ()
 
Subscriptoroperator= (const Subscriptor &)
 
Subscriptoroperator= (Subscriptor &&) noexcept
 
void subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const
 
void unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const
 
unsigned int n_subscriptions () const
 
template<typename StreamType >
void list_subscribers (StreamType &stream) 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< typename VectorType::value_type > &diagonal, const std::vector< typename VectorType::value_type > &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(typename VectorType::value_type, typename VectorType::value_type)> 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 SolverBase< 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), StateCombineriteration_status
 

Additional Inherited Members

- Static Public Member Functions inherited from Subscriptor
static ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)
 
static ::ExceptionBaseExcNoSubscriber (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 96 of file solver_cg.h.

Member Typedef Documentation

◆ size_type

template<typename VectorType = Vector<double>>
using SolverCG< VectorType >::size_type = types::global_dof_index

Declare type for container size.

Definition at line 102 of file solver_cg.h.

Constructor & Destructor Documentation

◆ SolverCG() [1/2]

template<typename VectorType = Vector<double>>
SolverCG< VectorType >::SolverCG ( SolverControl cn,
VectorMemory< VectorType > &  mem,
const AdditionalData data = AdditionalData() 
)

Constructor.

◆ SolverCG() [2/2]

template<typename VectorType = Vector<double>>
SolverCG< VectorType >::SolverCG ( SolverControl cn,
const AdditionalData data = AdditionalData() 
)

Constructor. Use an object of type GrowingVectorMemory as a default to allocate memory.

◆ ~SolverCG()

template<typename VectorType = Vector<double>>
virtual SolverCG< VectorType >::~SolverCG ( )
overridevirtualdefault

Virtual destructor.

Member Function Documentation

◆ solve()

template<typename VectorType = Vector<double>>
template<typename MatrixType , typename PreconditionerType >
void SolverCG< VectorType >::solve ( const MatrixType &  A,
VectorType &  x,
const VectorType &  b,
const PreconditionerType &  preconditioner 
)

Solve the linear system \(Ax=b\) for x.

◆ connect_coefficients_slot()

template<typename VectorType = Vector<double>>
boost::signals2::connection SolverCG< VectorType >::connect_coefficients_slot ( const std::function< void(typename VectorType::value_type, typename VectorType::value_type)> &  slot)

Connect a slot to retrieve the CG coefficients. The slot will be called with alpha as the first argument and with beta as the second argument, where alpha and beta follow the notation in Y. Saad: "Iterative methods for Sparse Linear Systems", section 6.7. Called once per iteration

◆ connect_condition_number_slot()

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

Connect a slot to retrieve the estimated condition number. Called on each iteration if every_iteration=true, otherwise called once when iterations are ended (i.e., either because convergence has been achieved, or because divergence has been detected).

◆ connect_eigenvalues_slot()

template<typename VectorType = Vector<double>>
boost::signals2::connection SolverCG< VectorType >::connect_eigenvalues_slot ( const std::function< void(const std::vector< double > &)> &  slot,
const bool  every_iteration = false 
)

Connect a slot to retrieve the estimated eigenvalues. Called on each iteration if every_iteration=true, otherwise called once when iterations are ended (i.e., either because convergence has been achieved, or because divergence has been detected).

◆ print_vectors()

template<typename VectorType = Vector<double>>
virtual void SolverCG< VectorType >::print_vectors ( const unsigned int  step,
const VectorType &  x,
const VectorType &  r,
const VectorType &  d 
) const
protectedvirtual

Interface for derived class. This function gets the current iteration vector, the residual and the update vector in each step. It can be used for graphical output of the convergence history.

◆ compute_eigs_and_cond()

template<typename VectorType = Vector<double>>
static void SolverCG< VectorType >::compute_eigs_and_cond ( const std::vector< typename VectorType::value_type > &  diagonal,
const std::vector< typename VectorType::value_type > &  offdiagonal,
const boost::signals2::signal< void(const std::vector< double > &)> &  eigenvalues_signal,
const boost::signals2::signal< void(double)> &  cond_signal 
)
staticprotected

Estimates the eigenvalues from diagonal and offdiagonal. Uses these estimate to compute the condition number. Calls the signals eigenvalues_signal and cond_signal with these estimates as arguments.

Member Data Documentation

◆ additional_data

template<typename VectorType = Vector<double>>
AdditionalData SolverCG< VectorType >::additional_data
protected

Additional parameters.

Definition at line 200 of file solver_cg.h.

◆ coefficients_signal

template<typename VectorType = Vector<double>>
boost::signals2::signal<void(typename VectorType::value_type, typename VectorType::value_type)> SolverCG< VectorType >::coefficients_signal
protected

Signal used to retrieve the CG coefficients. Called on each iteration.

Definition at line 207 of file solver_cg.h.

◆ condition_number_signal

template<typename VectorType = Vector<double>>
boost::signals2::signal<void(double)> SolverCG< VectorType >::condition_number_signal
protected

Signal used to retrieve the estimated condition number. Called once when all iterations are ended.

Definition at line 213 of file solver_cg.h.

◆ all_condition_numbers_signal

template<typename VectorType = Vector<double>>
boost::signals2::signal<void(double)> SolverCG< VectorType >::all_condition_numbers_signal
protected

Signal used to retrieve the estimated condition numbers. Called on each iteration.

Definition at line 219 of file solver_cg.h.

◆ eigenvalues_signal

template<typename VectorType = Vector<double>>
boost::signals2::signal<void(const std::vector<double> &)> SolverCG< VectorType >::eigenvalues_signal
protected

Signal used to retrieve the estimated eigenvalues. Called once when all iterations are ended.

Definition at line 225 of file solver_cg.h.

◆ all_eigenvalues_signal

template<typename VectorType = Vector<double>>
boost::signals2::signal<void(const std::vector<double> &)> SolverCG< VectorType >::all_eigenvalues_signal
protected

Signal used to retrieve the estimated eigenvalues. Called on each iteration.

Definition at line 232 of file solver_cg.h.


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