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 () |
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.
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.
SolverCG< VECTOR >::SolverCG | ( | SolverControl & | cn, | |
VectorMemory< VECTOR > & | mem, | |||
const AdditionalData & | data = AdditionalData() | |||
) |
Constructor.
SolverCG< VECTOR >::SolverCG | ( | SolverControl & | cn, | |
const AdditionalData & | data = AdditionalData() | |||
) |
Constructor. Use an object of type GrowingVectorMemory as a default to allocate memory.
Virtual destructor.
void SolverCG< VECTOR >::solve | ( | const MATRIX & | A, | |
VECTOR & | x, | |||
const VECTOR & | b, | |||
const PRECONDITIONER & | precondition | |||
) | [inline] |
Solve the linear system for x.
Referenced by SolverSelector< VECTOR >::solve().
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.
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.
void SolverCG< VECTOR >::cleanup | ( | ) | [private] |
Temporary vectors, allocated through the VectorMemory
object at the start of the actual solution process and deallocated at the end.
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.
AdditionalData SolverCG< VECTOR >::additional_data [protected] |
Additional parameters.