Reference documentation for deal.II version Git 3041c91f43 2021-03-05 14:58:09 +0100
SolverCG< VectorType > Class Template Reference

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

Inheritance diagram for SolverCG< VectorType >:
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## Public Types

using size_type = types::global_dof_index

using vector_type = VectorType

## Public Member Functions

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)

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

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

Subscriptor functionality

Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.

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

## Static Public Member Functions

static ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBaseExcNoSubscriber (std::string arg1, std::string arg2)

## 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

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

GrowingVectorMemory< VectorTypestatic_vector_memory

VectorMemory< VectorType > & memory

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

## 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.

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.

Definition at line 95 of file solver_cg.h.

## ◆ size_type

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

Declare type for container size.

Definition at line 101 of file solver_cg.h.

## ◆ vector_type

template<class VectorType = Vector<double>>
 using SolverBase< VectorType >::vector_type = VectorType
inherited

An alias for the underlying vector type

Definition at line 347 of file solver.h.

## ◆ 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.

## ◆ 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.

## ◆ connect()

template<class VectorType >
 boost::signals2::connection SolverBase< VectorType >::connect ( const std::function< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType ¤t_iterate)> & slot )
inlineinherited

Connect a function object that will be called periodically within iterative solvers. This function is used to attach monitors to iterative solvers, either to determine when convergence has happened, or simply to observe the progress of an iteration. See the documentation of this class for more information.

Parameters
 slot A function object specified here will, with each call, receive the number of the current iteration, the value that is used to check for convergence (typically the residual of the current iterate with respect to the linear system to be solved) and the currently best available guess for the current iterate. Note that some solvers do not update the approximate solution in every iteration but only after convergence or failure has been determined (GMRES is an example); in such cases, the vector passed as the last argument to the signal is simply the best approximate at the time the signal is called, but not the vector that will be returned if the signal's return value indicates that the iteration should be terminated. The function object must return a SolverControl::State value that indicates whether the iteration should continue, has failed, or has succeeded. The results of all connected functions will then be combined to determine what should happen with the iteration.
Returns
A connection object that represents the connection from the signal to the function object. It can be used to disconnect the function object again from the signal. See the documentation of the BOOST Signals2 library for more information on connection management.

Definition at line 554 of file solver.h.

## ◆ subscribe()

 void Subscriptor::subscribe ( std::atomic< bool > *const validity, const std::string & identifier = "" ) const
inherited

Subscribes a user of the object by storing the pointer validity. The subscriber may be identified by text supplied as identifier.

Definition at line 136 of file subscriptor.cc.

## ◆ unsubscribe()

 void Subscriptor::unsubscribe ( std::atomic< bool > *const validity, const std::string & identifier = "" ) const
inherited

Unsubscribes a user from the object.

Note
The identifier and the validity pointer must be the same as the one supplied to subscribe().

Definition at line 156 of file subscriptor.cc.

## ◆ n_subscriptions()

 unsigned int Subscriptor::n_subscriptions ( ) const
inlineinherited

Return the present number of subscriptions to this object. This allows to use this class for reference counted lifetime determination where the last one to unsubscribe also deletes the object.

Definition at line 301 of file subscriptor.h.

## ◆ list_subscribers() [1/2]

template<typename StreamType >
 void Subscriptor::list_subscribers ( StreamType & stream ) const
inlineinherited

List the subscribers to the input stream.

Definition at line 318 of file subscriptor.h.

## ◆ list_subscribers() [2/2]

 void Subscriptor::list_subscribers ( ) const
inherited

List the subscribers to deallog.

Definition at line 204 of file subscriptor.cc.

## ◆ serialize()

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

Read or write the data of this object to or from a stream for the purpose of serialization using the BOOST serialization library.

This function does not actually serialize any of the member variables of this class. The reason is that what this class stores is only who subscribes to this object, but who does so at the time of storing the contents of this object does not necessarily have anything to do with who subscribes to the object when it is restored. Consequently, we do not want to overwrite the subscribers at the time of restoring, and then there is no reason to write the subscribers out in the first place.

Definition at line 310 of file subscriptor.h.

## Member Data Documentation

template<typename VectorType = Vector<double>>
protected

Definition at line 199 of file solver_cg.h.

## ◆ coefficients_signal

template<typename VectorType = Vector<double>>
 boost::signals2::signal SolverCG< VectorType >::coefficients_signal
protected

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

Definition at line 206 of file solver_cg.h.

## ◆ condition_number_signal

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

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

Definition at line 212 of file solver_cg.h.

## ◆ all_condition_numbers_signal

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

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

Definition at line 218 of file solver_cg.h.

## ◆ eigenvalues_signal

template<typename VectorType = Vector<double>>
 boost::signals2::signal &)> SolverCG< VectorType >::eigenvalues_signal
protected

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

Definition at line 224 of file solver_cg.h.

## ◆ all_eigenvalues_signal

template<typename VectorType = Vector<double>>
 boost::signals2::signal &)> SolverCG< VectorType >::all_eigenvalues_signal
protected

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

Definition at line 231 of file solver_cg.h.

## ◆ static_vector_memory

template<class VectorType = Vector<double>>
 GrowingVectorMemory SolverBase< VectorType >::static_vector_memory
mutableprotectedinherited

A static vector memory object to be used whenever no such object has been given to the constructor.

Definition at line 415 of file solver.h.

## ◆ memory

template<class VectorType = Vector<double>>
 VectorMemory& SolverBase< VectorType >::memory
protectedinherited

A reference to an object that provides memory for auxiliary vectors.

Definition at line 420 of file solver.h.

## ◆ iteration_status

template<class VectorType = Vector<double>>
 boost::signals2::signal< SolverControl::State(const unsigned int iteration, const double check_value, const VectorType & current_iterate), StateCombiner> SolverBase< VectorType >::iteration_status
protectedinherited

A signal that iterative solvers can execute at the end of every iteration (or in an otherwise periodic fashion) to find out whether we should continue iterating or not. The signal may call one or more slots that each will make this determination by themselves, and the result over all slots (function calls) will be determined by the StateCombiner object.

The arguments passed to the signal are (i) the number of the current iteration; (ii) the value that is used to determine convergence (oftentimes the residual, but in other cases other quantities may be used as long as they converge to zero as the iterate approaches the solution of the linear system); and (iii) a vector that corresponds to the current best guess for the solution at the point where the signal is called. Note that some solvers do not update the approximate solution in every iteration but only after convergence or failure has been determined (GMRES is an example); in such cases, the vector passed as the last argument to the signal is simply the best approximate at the time the signal is called, but not the vector that will be returned if the signal's return value indicates that the iteration should be terminated.

Definition at line 471 of file solver.h.

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