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Solving eigenvalue problems by implicit decomposition. (English) Zbl 0734.65028

Three methods are presented for treating eigenvalue problems of the type \(Kx+\lambda Mx=0,\) where K and M can be unsymmetric, indefinite and even singular matrices of high order. The basic idea consists in applying a decomposition of the given problem in subproblems that can be treated in parallel and that exploit the sparsity of the matrices.
However, the first method is just a simplified version of coordinate relaxation, as introduced by Faddeev and its block variant. The second method is an extension of the first, and the third method is said to be adapted for the computation of multiple eigenpairs.
First of all, it is known, that these methods do have bad convergence properties. Secondly, the presented methods require that the starting approximations must be chosen within the (probably quite small) convergence region of an eigenpair. Such a restriction certainly does not fulfil the claimed property of robustness of the proposed methods.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
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References:

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