# On the Convergence of Ritz Values, Ritz Vectors, and Refined Ritz Vectors\symbolmark

 dc.contributor.author Jai, Zhongxiao en_US dc.contributor.author Stewart, G. W. en_US dc.date.accessioned 2004-05-31T22:55:48Z dc.date.available 2004-05-31T22:55:48Z dc.date.created 1999-01 en_US dc.date.issued 1999-01-29 en_US dc.identifier.uri http://hdl.handle.net/1903/993 dc.description.abstract This paper concerns the Rayleigh--Ritz method for computing an approximation to an eigenpair $(\lambda, x)$ of a non-Hermitian matrix $A$. Given a subspace $\clw$ that contains an approximation to $x$, this method returns an approximation $(\mu, \tilde x)$ to $(\lambda, x)$. We establish four convergence results that hold as the deviation $\epsilon$ of $x$ from $\clw$ approaches zero. First, the Ritz value $\mu$ converges to $\lambda$. Second, if the residual $A\tilde x-\mu\tilde x$ approaches zero, then the Ritz vector $\tilde x$ converges to $x$. Third, we give a condition on the eigenvalues of the Rayleigh quotient from which the Ritz pair is computed that insures convergence of the Ritz vector. Finally, we show that certain unconditionally. (Also cross-referenced as UMIACS-TR-99-08) en_US dc.format.extent 150796 bytes dc.format.mimetype application/postscript dc.language.iso en_US dc.relation.ispartofseries UM Computer Science Department; CS-TR-3986 en_US dc.relation.ispartofseries UMIACS; UMIACS-TR-99-08 en_US dc.title On the Convergence of Ritz Values, Ritz Vectors, and Refined Ritz Vectors\symbolmark en_US dc.type Technical Report en_US dc.relation.isAvailableAt Digital Repository at the University of Maryland en_US dc.relation.isAvailableAt University of Maryland (College Park, Md.) en_US dc.relation.isAvailableAt Tech Reports in Computer Science and Engineering en_US dc.relation.isAvailableAt UMIACS Technical Reports en_US
﻿