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1. Introduction. It is the main purpose of this paper to compare the mean rates of convergence of two well-known schemes for solving self-adjoint partial difference equations of elliptic type: the Young-Frankel [6,1] successive overrelaxation method, and the semi-iterative Chebyshev polynomial method as described by Lanczos [2, p.42], Stiefel [3], and others. More generally, the analysis is applicable to any matrix equation of the form...