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Acceleration of relaxation methods for non-Hermitian linear systems. (English) Zbl 0757.65032

When solving a general linear system \(Ax=b\), it is possible to apply an extension of the already classical relaxation theory by considering what the authors define as the Hermitian and skew-Hermitian splittings of \(A\); two families of equivalent systems are thus derived. These systems can be subject to \(k\)-step methods in order to obtain accelerated iterative methods.
Since the available information concerning the spectrum of the Hermitian and skew-Hermitian parts of \(A\) is of local nature, the relaxation factors to be chosen should not have as its main objective the production of the fastest relaxed iterations; instead, those factors should be looked for that lead to the most efficient accelerations.
For the Hermitian splitting coupled with a Chebyshev acceleration the factior 1 is shown to be optimal. A hybrid method is proposed for the skew-Hermitian approach. The paper also contains several enlightening examples.

MSC:

65F10 Iterative numerical methods for linear systems
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