On the Optimization of the Classical Iterative Schemes for the Solution of Complex Singular Linear Systems

For the numerical solution of a class of Complex Singular Linear Systems $Ax = b$, with $\det ( A ) = 0$ and b in the range of A, the generalized iterative methods of Extrapolated Jacobi (JOR) and of Successive Overrelaxation (SOR), first introduced by Buoni and Varga, are considered. Under some basic assumptions the various parameters of the optimal Generalized JOR and SOR schemes are determined through formulas given by means of specific algorithms which are proposed. A number of numerical examples are also presented to show how one can apply the algorithms and determine, subsequently, the optimal parameters which make the corresponding schemes to semiconverge as fast as possible.