laspack.tgz: laspack/html/node24.html

"Fossies" - the Fresh Open Source Software Archive

Member "laspack/html/node24.html" (13 Aug 1995, 6098 Bytes) of package /linux/privat/old/laspack.tgz:

Caution: In this restricted "Fossies" environment the current HTML page may not be correctly presentated and may have some non-functional links. You can here alternatively try to browse the pure source code or just view or download the uninterpreted raw source code. If the rendering is insufficient you may try to find and view the page on the project site itself.

Next: QMATRIX(3LAS) Up: Manual Pages Previous: OPERATS(3LAS)

PRECOND(3LAS)

NAME

JacobiPrecond, SSORPrecond, ILUPrecond -- pre-defined preconditioners

SYNOPSIS

#include <laspack/precond.h>

typedef Vector *(*PrecondProcType)(QMatrix *, Vector *, Vector *, double)

Vector *JacobiPrecond(QMatrix *A, Vector *y, Vector *c, double Omega); 
Vector *SSORPrecond(QMatrix *A, Vector *y, Vector *c, double Omega); 
Vector *ILUPrecond(QMatrix *A, Vector *y, Vector *c, double Omega);

DESCRIPTION

In LASPack , three preconditioners are currently available. They will be applied to the system

    W y = c

which arises during the solution of the preconditioned systems of equations

    W^{-1} A x = W^{-1} b.

The matrix W is in all three cases derived from the matrix A . The parameter Omega is used as relaxation parameter.

The procedure JacobiPrecond performs the preconditioning by means of weighted diagonal of the original matrix

    W = 1 / Omega . diag(A)

so that

    y = Omega diag(A)^{-1} c.

For preconditioning by the Symmetric Successive Over-relaxation Method, the matrix A is decomposed in the diagonal D, the lower triangular part L, and the upper triangular part U :

    A = D + L + U.

The procedure SSORPrecond uses then

    W = 1 / (2 - Omega) . (D / Omega + L) (D / Omega)^{-1} (D / Omega + U).

Hence follows

    y = (2 - Omega) / Omega . (D / Omega + U)^{-1} D (D / Omega + L)^{-1} c.

The usage of ILUPrecond assumes an incomplete factorization of the matrix A in the form

    A = W + R = (D + L) D^{-1} (D + U) + R

Here D, U, and L are certain diagonal, upper, and lower triangular matrices, respectively, the remainder matrix R contains fill elements, which have been ignored during the factorization process.

The matrix W can be inverted exactly so that

    y = (D + U)^{-1} D (D + L)^{-1} c.

All three preconditioners are applicable for both, symmetric and non-symmetric matrices. For symmetric matrices, the symmetry condition L = U^T is taken into account.

REFERENCES

For description and theoretical foundation of above preconditioners see e.g.:

W. Hackbusch: Iterative Solution of Large Sparse Systems of Equations, Springer-Verlag, Berlin, 1994.

FILES

precond.h ... header file
precond.c ... source file

EXAMPLES

Preconditioners as well as iterative solvers are defined in LASPack by means of matrix-vector operations, especially the procedures Mul_QV and MulInv_QV from module OPERATS. This keep they independently on the matrix storage format. The Jacobi preconditioner is e.g. implemented by the following one-line statement:

Vector *JacobiPrecond(QMatrix *A, Vector *y, Vector *c, double Omega)
{
    Q_Lock(A);
    V_Lock(y);
    V_Lock(c);

    Asgn_VV(y, Mul_SV(Omega, MulInv_QV(Diag_Q(A), c)));

    Q_Unlock(A);
    V_Unlock(y);
    V_Unlock(c);

    return(y);
}