Further results on regular splittings and their applications to some iterative methods

Abstract

In this paper we present a general theorem which allows the comparison of the spectral radius of the iteration matrices of some iterative methods, obtained from different regular splittings of the matrix A of Ax = b. Thus, when A is a non-singular M-matrix, not necessary irreducible, we can prove that the Gauss–Seidel method is faster than the (SOR) iterative method (). In these conditions, we can choose convenient values for the parameters of the Generalised Accelerated Overrelaxation method (GAOR) and get faster results. Thus, with 0 < r 2, w 2 ≤ 1 the (GAOR) method is faster than when we approximate the solution of Ax=b with 0 < r 1 w 1 ≤ 1 and and with . We also prove that, under some assumptions, the GAOR method can be faster than the AOR method and, under other assumptions, the reverse is also true which improves and generalises the results obtained in [3].