The preconditioned simultaneous displacement method (PSD method) for elliptic difference equations

Abstract

This paper introduces the Preconditioned Simultaneous Displacement iterative method (PSD method) in a new “computable” form for the numerical solution of linear systems of the form Au=b, where the matrix A is large and sparse. The convergence properties of the method are analysed under certain assumptions on the matrix A. Moreover, “good” values (near the optimum) for the involved parameters are determined in terms of bounds on the eigenvalues of certain matrices. Bounds on the reciprocal rate of convergence of the PSD method are also given. The method is shown to be superior over the well known Symmetric Successive Overrelaxation method (SSOR method) (at the optimum stage PSD is shown to converge approximately two times faster than SSOR) and in certain cases over the Successive Overrelaxation method (SOR method).