The modified preconditioned simultaneous displacement (MPSD) method

In [4] it is shown that the Preconditioned Simultaneous Displacement (PSD) method when applied to σ2-ordered systems of linear equations may be preferred over the Successive Overrelaxation (SOR) method in certain cases whereas it is undoubtedly superior over the symmetric SOR (SSOR) method. In this paper, we define a generalised form of PSD and the produced scheme, the Modified PSD (MPSD) method is considered for the solution of the linear system Au=b when A possesses a special form (σ1-ordered). The analysis of convergence and the determination of optimum values for the involved parameters is presented. It is shown that if MPSD is combined with a work reduction scheme, then it may be considered competitive with SOR as the two methods possess the same rate of convergence at the optimum stage. Finally, it is proven that under these conditions PSD possesses a rate of convergence which is asymptotically twice as fast as SSOR, a result which is also valid for σ2-ordered systems.