SolverCG< VECTOR > Class Template Reference
[Linear solver classes]

Inheritance diagram for SolverCG< VECTOR >:
Inheritance graph
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List of all members.

Classes

struct  AdditionalData

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)

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
VECTOR * VAp
double res2
AdditionalData additional_data

Private Member Functions

void cleanup ()

Detailed Description

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

Preconditioned cg method for symmetric positive definite matrices.

For the requirements on matrices and vectors in order to work with this class, see the documentation of the Solver base class.

Like all other solver classes, this class has a local structure called AdditionalData which is used to pass additional parameters to the solver, like damping parameters or the number of temporary vectors. For this class, there is a switch allowing for additional output for the computation of eigenvalues of the matrix.

Eigenvalue computation

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

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.

This version of CG is taken from Braess: "Finite Elements". It requires a symmetric preconditioner, i.e. SOR is not feasible.

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

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 
) [inline]

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

Referenced by SolverSelector< VECTOR >::solve().

template<class VECTOR = Vector<double>>
virtual double SolverCG< VECTOR >::criterion (  )  [protected, virtual]

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 [protected, virtual]

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.

template<class VECTOR = Vector<double>>
void SolverCG< VECTOR >::cleanup (  )  [private]

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.

template<class VECTOR = Vector<double>>
VECTOR* SolverCG< VECTOR >::Vp [protected]
template<class VECTOR = Vector<double>>
VECTOR* SolverCG< VECTOR >::Vz [protected]
template<class VECTOR = Vector<double>>
VECTOR* SolverCG< VECTOR >::VAp [protected]
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.

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

Additional parameters.


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

deal.II documentation generated on Mon Nov 23 22:58:05 2009 by doxygen 1.6.1