Reference documentation for deal.II version Git ef01321 2014-09-18 06:20:48 -0500
SolverCG< VECTOR > Class Template Reference

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

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

typedef types::global_dof_index size_type

## Public Member Functions

virtual ~SolverCG ()

template<class MATRIX , class PRECONDITIONER >
void solve (const MATRIX &A, VECTOR &x, const VECTOR &b, const PRECONDITIONER &precondition)

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

Solver (SolverControl &solver_control)

SolverControlcontrol () const

Public Member Functions inherited from Subscriptor
Subscriptor ()

Subscriptor (const Subscriptor &)

virtual ~Subscriptor ()

Subscriptoroperator= (const Subscriptor &)

void subscribe (const char *identifier=0) const

void unsubscribe (const char *identifier=0) const

unsigned int n_subscriptions () const

void list_subscribers () const

DeclException3 (ExcInUse, int, char *, std::string &,<< "Object of class "<< arg2<< " is still used by "<< arg1<< " other objects.\n"<< "(Additional information: "<< arg3<< ")\n"<< "Note the entry in the Frequently Asked Questions of "<< "deal.II (linked to from http://www.dealii.org/) for "<< "more information on what this error means.")

DeclException2 (ExcNoSubscriber, char *, char *,<< "No subscriber with identifier \""<< arg2<< "\" did subscribe to this object of class "<< arg1)

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

## Protected Member Functions

virtual double criterion ()

virtual void print_vectors (const unsigned int step, const VECTOR &x, const VECTOR &r, const VECTOR &d) const

## Protected Attributes

VECTOR * Vr

VECTOR * Vp

VECTOR * Vz

double res2

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

SolverControlcntrl

VectorMemory< VECTOR > & memory

void cleanup ()

## Detailed Description

### template<class VECTOR = Vector<double>> class SolverCG< VECTOR >

Preconditioned cg method for symmetric positive definite matrices. 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>.

Like all other solver classes, this class has a local structure called AdditionalData which is used to pass additional parameters to the solver. For this class, there is (among other things) a switch allowing for additional output for the computation of eigenvalues of the matrix.

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.

With the coefficients alpha and beta written to the log file if AdditionalData::log_coefficients = true, the matrix T_m after m steps is 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 Y. Saad: "Iterative methods for Sparse Linear Systems", section 6.7.3 for details.

Definition at line 85 of file solver_cg.h.

## Member Typedef Documentation

template<class VECTOR = Vector<double>>
 typedef types::global_dof_index SolverCG< VECTOR >::size_type

Declare type for container size.

Definition at line 91 of file solver_cg.h.

## Constructor & Destructor Documentation

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

Constructor.

template<class VECTOR = Vector<double>>
 SolverCG< VECTOR >::SolverCG ( SolverControl & cn, const AdditionalData & data = AdditionalData() )

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

template<class VECTOR = Vector<double>>
 virtual SolverCG< VECTOR >::~SolverCG ( )
virtual

Virtual destructor.

## Member Function Documentation

template<class VECTOR = Vector<double>>
template<class MATRIX , class PRECONDITIONER >
 void SolverCG< VECTOR >::solve ( const MATRIX & A, VECTOR & x, const VECTOR & b, const PRECONDITIONER & precondition )

Solve the linear system $$Ax=b$$ for x.

template<class VECTOR = Vector<double>>
 virtual double SolverCG< VECTOR >::criterion ( )
protectedvirtual

Implementation of the computation of the norm of the residual. This can be replaced by a more problem oriented functional in a derived class.

template<class VECTOR = Vector<double>>
 virtual void SolverCG< VECTOR >::print_vectors ( const unsigned int step, const VECTOR & x, const VECTOR & r, const VECTOR & 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 a graphical output of the convergence history.

## Member Data Documentation

template<class VECTOR = Vector<double>>
 VECTOR* SolverCG< VECTOR >::Vr
protected

Temporary vectors, allocated through the VectorMemory object at the start of the actual solution process and deallocated at the end.

Definition at line 203 of file solver_cg.h.

template<class VECTOR = Vector<double>>
 double SolverCG< VECTOR >::res2
protected

Within the iteration loop, the square of the residual vector is stored in this variable. The function criterion uses this variable to compute the convergence value, which in this class is the norm of the residual vector and thus the square root of the res2 value.

Definition at line 217 of file solver_cg.h.

template<class VECTOR = Vector<double>>