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Inexact Rayleigh quotient-type methods for eigenvalue computations
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.ORCID iD: 0000-0003-2281-856X
2002 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, Vol. 42, no 1, 159-182 p.Article in journal (Refereed) Published
Abstract [en]

We consider the computation of an eigenvalue and corresponding eigenvector of a Hermitian positive definite matrix A is an element of C-nxn, assuming that good approximations of the wanted eigenpair are already available, as may be the case in applications such as structural mechanics. We analyze efficient implementations of inexact Rayleigh quotient-type methods, which involve the approximate solution of a linear system at each iteration by means of the Conjugate Residuals method. We show that the inexact version of the classical Rayleigh quotient iteration is mathematically equivalent to a Newton approach. New insightful bounds relating the inner and outer recurrences are derived. In particular, we show that even if in the inner iterations the norm of the residual for the linear system decreases very slowly, the eigenvalue residual is reduced substantially. Based on the theoretical results, we examine stopping criteria for the inner iteration. We also discuss and motivate a preconditioning strategy for the inner iteration in order to further accelerate the convergence. Numerical experiments illustrate the analysis.

Place, publisher, year, edition, pages
2002. Vol. 42, no 1, 159-182 p.
Keyword [en]
eigenvalue approximation, iterative methods, Newton method, inexact Rayleigh quotient iteration
National Category
Engineering and Technology
URN: urn:nbn:se:liu:diva-48163OAI: diva2:269059
Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2013-08-30

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