Reference documentation for deal.II version Git c74a91e 2014-12-19 05:57:33 -0600
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SolverCG< VECTOR > Class Template Reference

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

Inheritance diagram for SolverCG< VECTOR >:


struct  AdditionalData

Public Types

typedef types::global_dof_index size_type

Public Member Functions

 SolverCG (SolverControl &cn, VectorMemory< VECTOR > &mem, const AdditionalData &data=AdditionalData())
 SolverCG (SolverControl &cn, const AdditionalData &data=AdditionalData())
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)
boost::signals2::connection connect (const std_cxx11::function< SolverControl::State(const unsigned int iteration, const doublecheck_value, const VECTOR &current_iterate)> &slot)
- 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 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

double res2
AdditionalData additional_data
- Protected Attributes inherited from Solver< VECTOR >
GrowingVectorMemory< VECTOR > static_vector_memory
VectorMemory< VECTOR > & memory
< SolverControl::State(const
unsigned int iteration, const
double check_value, const
VECTOR &current_iterate),

Private Member Functions

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.

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

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.

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

Definition at line 91 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 97 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() 


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 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 ( )

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

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>>

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

Definition at line 190 of file solver_cg.h.

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

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 200 of file solver_cg.h.

template<class VECTOR = Vector<double>>
AdditionalData SolverCG< VECTOR >::additional_data

Additional parameters.

Definition at line 205 of file solver_cg.h.

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