Optimal block iterative schemes for certain large, sparse, and nonsymmetric linear systems

Abstract

The problem of determining fast iterative solutions of certain large, sparse, and nonsymmetric linear systems, arising in applications, is addressed here. Several iterative schemes, from the accelerated overrelaxation family, are considered. Different geometrical algorithms are used for the explicit determination of the optimal factors. Direct comparisons of the spectral radii of the resulting optimal schemes reveal that the optimal extrapolated accelerated Gauss-Seidel (EAGS) is always asymptotically faster than the optimal successive overrelaxation, while the optimal EAGS and extrapolated Gauss-Seidel strongly compete. Application of the collocation method on simple boundary-value problems is used to demonstrate our results numerically.