×

Explicit overconditioning of systems of linear algebraic equations with dense matrices. (Russian) Zbl 0621.65041

For linear systems of equations \(Ax=b\) with full nonsymmetric nondiagonal dominant matrix of great order an explicit preconditioning G for the best convergence of an iterative method, in this case ORTHOMIN(k), is suggested. The structure of the preconditioning matrix G depends on the module of the elements of the matrix A. Three types of the strategy are given:
1) one-step strategy for the non-zero elements of \(G_ 1\); 2) adaptive improving of G, i.e. another \(G_ 2\) is taken or \(G_ 2G_ 1A\) is used; 3) two-step preconditioning, where only some of the rows of the matrix \(G_ 2\) are different from the rows of the identity matrix. The last strategy seems to be the best one. One example for the Laplace equation in 3 dimensions with boundary conditions is shown for the demonstration.

MSC:

65F35 Numerical computation of matrix norms, conditioning, scaling
65F10 Iterative numerical methods for linear systems
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
PDFBibTeX XMLCite
Full Text: EuDML