Luo, Jenn-Ching Solving eigenvalue problems by implicit decomposition. (English) Zbl 0734.65028 Numer. Methods Partial Differ. Equations 7, No. 2, 113-145 (1991). 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. Reviewer: H.R.Schwarz (Zürich) Cited in 1 ReviewCited in 1 Document MSC: 65F15 Numerical computation of eigenvalues and eigenvectors of matrices Keywords:generalized eigenvalue problem; decomposition; coordinate relaxation; multiple eigenpairs; convergence PDFBibTeX XMLCite \textit{J.-C. Luo}, Numer. Methods Partial Differ. Equations 7, No. 2, 113--145 (1991; Zbl 0734.65028) Full Text: DOI References: [1] and , ”Large scale eigenvalue problems,” Proceedings of the IBM Europe Institute Workshop on Large Scale Eigenvalue Problems, Oberlech, Austria, July 8-12, 1985, North-Holland, 1986. [2] and , Lanczos Algorithms for Large Symmetric Eigenvalue Computations. Vol. I. Theory, Birkhäuser, 1985. [3] The Algebraic Eigenvalue Problem, Oxford University Press, UK, 1988. · Zbl 0626.65029 [4] Luo, Comput. & Math. with Appl. 21 (1991) · Zbl 0723.65095 · doi:10.1016/0898-1221(91)90053-7 [5] ”Parallel algorithms for the finite element method, ” Ph.D. Thesis, Columbia University, 1988. [6] Luo, Mech. Res. Commun. 16 (1989) [7] Luo, Computers and Structures 35 (1990) [8] Luo, Comput. Meth. Appl. Mech. Eng. 84 (1990) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.