Localization Criteria and Containment for Rayleigh Quotient Iteration

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Abstract
Rayleigh quotient iteration can often yield an eigenvalue-eigenvector pair of a positive-definite Hermitian problem in a very short time. The primary hindrance associated with its use as a regular computational tool lies with the difficulty of identifying and selecting the final regions of convergence. In this paper rigorous, accessible criteria for localizing Rayleigh quotient iteration to prespecified intervals of the spectrum are provided, as well as extensions to situations where only partial spectral information is available. An application for finding partial eigensolutions of symmetric tridiagonal matrices is given with results that compare very favorably with the EISPACK routine TSTURM.