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We consider the successive overrelaxation (SOR) and related methods for the linear system (1.1) with the coefficient matrix A being p-cyclic. Li [1989], Pierce, Hadjidimos and Plemmons [1990] have shown that when the SOR method is used it is best to partition the p-cyclic matrix A into a 2-cyclic form under some conditions. Recently, another proof of the result was given by Eiermann, Niethammer and Ruttan [1990]. In this paper, firstly an alternative proof of their result is given, which is simpler and more straightforward. Secondly, an quantitative analysis is given. It is shown that the SOR method applied to (1.1) with A p-cyclic requires up to 40% more iterations in some cases than the SOR scheme applied to (1.1) with A partitioned as 2-cyclic, called the SOR-2 scheme. Thirdly, we show that the SOR-2 scheme has the same asymptotic rate of convergence as the Chebyshev acceleration of the Gauss-Seidel (GS) method. Finally, the properties of the symmetric SOR (SSOR) method are investigated numerically when the p-cyclic system is repartitioned into k-cyclic (2≤k≤p) form.

A direct-iterative method is developed for the numerical solution of the heat conduction equation having period T in time. Numerical results have shown that the method is better than the iterative method of Tee [1] for any T and the direct method of Osborne [2] for T≥ 10-3.