ABSTRACT In this paper we extend the theory of a block preconditioned SOR method studied by Hezari, Edalaptour, and Salkuyeh (2015) for the solution of indefinite complex linear systems. In particular, we consider the case where the key matrix has real eigenvalues which lie in and not only in as is assumed up to the present. Under this assumption we study the convergence behavior of the aforementioned iterative method and determine the optimum values of the involved parameters and the corresponding optimum spectral radius of its iteration matrix. Our theoretical analysis is based on the work by Niethammer (1979) and shows that the block preconditioned SOR method has, in most of the cases, at least an order of magnitude better rate of convergence over the rate of convergence of the block SOR method. Additional comparisons reveal that the block preconditioned SOR method compares favorably not only as a preconditioner to the GMRES method but also as a solution method to the recently proposed SNSS method 38, 34 for the numerical solution of indefinite complex linear systems. Finally, numerical results confirm our theoretical expectations.
Louka et al. (Sun,) studied this question.
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