Cluster robust error estimates for the Rayleigh-Ritz approximation II: Estimates for eigenvalues

Ovtchinnikov, E. 2006. Cluster robust error estimates for the Rayleigh-Ritz approximation II: Estimates for eigenvalues. Linear Algebra and its Applications. 415 (1), pp. 188-209.

TitleCluster robust error estimates for the Rayleigh-Ritz approximation II: Estimates for eigenvalues
AuthorsOvtchinnikov, E.
Abstract

This is the second part of a paper that deals with error estimates for the Rayleigh-Ritz approximations of the spectrum and invariant subspaces of a bounded Hermitian operator in a Hilbert or Euclidean space. This part addresses the approximation of eigenvalues. Two kinds of estimates are considered: (i) estimates for the eigenvalue errors via the best approximation errors for the corresponding invariant subspaces, and (ii) estimates for the same via the corresponding residuals. Estimates of these two kinds are needed for, respectively, the a priori and a posteriory error analysis of numerical methods for computing eigenvalues. The paper's major concern is to ensure that the estimates in question are accurate and 'cluster robust', i.e. are not adversely affected by the presence of clustered, i.e. closely situated eigenvalues among those of interest. The paper's main new results introduce estimates for clustered eigenvalues whereby not only the distances between eigenvalues in the cluster are not present but also the distances between the cluster and the rest of the spectrum appear in asymptotically insignificant terms only.

KeywordsSelf-adjoint eigenvalue problem, Rayleigh–Ritz method, a priori and a posteriori error estimates, Clustered eigenvalues
JournalLinear Algebra and its Applications
Journal citation415 (1), pp. 188-209
ISSN0024-3795
YearMay 2006
Digital Object Identifier (DOI)doi:10.1016/j.laa.2005.06.041
Publication dates
PublishedMay 2006

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