Cluster robust error estimates for the Rayleigh-Ritz approximation I: Estimates for invariant subspaces

Ovtchinnikov, E. 2006. Cluster robust error estimates for the Rayleigh-Ritz approximation I: Estimates for invariant subspaces. Linear Algebra and its Applications. 415 (1), pp. 167-187.

TitleCluster robust error estimates for the Rayleigh-Ritz approximation I: Estimates for invariant subspaces
AuthorsOvtchinnikov, E.
Abstract

This is the first part of a paper that deals with error estimates for the Rayleigh-Ritz approximations to the spectrum and invariant subspaces of a bounded Hermitian operator in a Hilbert or Euclidean space. This part addresses estimates for the angles between the invariant subspaces and their approximations via the corresponding best approximation errors and residuals and, for invariant subspaces corresponding to parts of the discrete spectrum, via eigenvalue errors. 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 in the spectrum. Available estimates of such kind are reviewed and new estimates are derived. The paper's main new results introduce estimates for invariant subspaces in which the operator may have 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, Invariant subspaces
JournalLinear Algebra and its Applications
Journal citation415 (1), pp. 167-187
ISSN0024-3795
YearMay 2006
Digital Object Identifier (DOI)doi:10.1016/j.laa.2005.06.040
Publication dates
PublishedMay 2006

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