Abstract The comparison of the asymptotic rates of convergence of two iteration matrices induced by two splitting of the same matrix has arisen in the work of many authors. In this article, some new comparison theorems for two nonnegative splittings (a splitting A = M − N is nonnegative if M −1 exist and M −1 N is nonnegative) are derived. They extend the known results in the literature. In addition, we also point out three incorrect conditions in a paper by Beauwens. Furthermore, we give some reasonable conditions ensuring the strict inequality between the asymptotic convergence rates. This also answers the open question which additional and appropriate conditions should be imposed on Miller-Neumann splittings to obtain strict inequality. Finally, some applications to a class of generalized AOR, SOR, and JOR iterative methods whose special cases imply block (also point) AOR, SOR, and JOR iterative methods for solving linear systems are discussed.
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