Clusters, preconditioners, convergence. (English) Zbl 0917.65032
The authors address the problem of clusterization of singular values. In this context, they present a technique that relates the existence of clusters of singular values to the existence of clusters of eigenvalues. This clusterization technique is based on the notion of regular preconditioners for a sequence of complex matrices. The technique is applied to the analysis of convergence properties of the preconditioned generalized minimal residual (GMRES) method. Furthermore, circulant preconditioners for Toeplitz matrices generated with \({\mathcal L}_2\) integrable functions are constructed by means of the clusterization technique.
Reviewer: J.R.Sendra (Madrid)
MSC:
| 65F10 | Iterative numerical methods for linear systems |
| 15A42 | Inequalities involving eigenvalues and eigenvectors |
| 65F35 | Numerical computation of matrix norms, conditioning, scaling |
Keywords:
generalized minimal residual method; GMRES; clusters of singular values; clusters of eigenvalues; convergence; circulant preconditioners; Toeplitz matricesReferences:
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