# Analysis of the Residual Arnoldi Method

 dc.contributor.author Lee, Che-Rung dc.contributor.author Stewart, G. W. dc.date.accessioned 2007-10-15T17:23:08Z dc.date.available 2007-10-15T17:23:08Z dc.date.issued 2007-10-15 dc.identifier.uri http://hdl.handle.net/1903/7428 dc.description.abstract The Arnoldi method generates a nested squences of orthonormal bases en $U_{1},U_{2}, \ldots$ by orthonormalizing $Au_{k}$ against $U_{k}$. Frequently these bases contain increasingly accurate approximations of eigenparis from the periphery of the spectrum of $A$. However, the convergence of these approximations stagnates if $u_{k}$ is contaminated by error. It has been observed that if one chooses a Rayleigh--Ritz approximation $(\mu_{k}, z_{k})$ to a chosen target eigenpair $(\lambda, x)$ and orthonormalizes the residual $Az_{k - }\mu_{k} z_{k}$, the approximations to $x$ (but not the other eigenvectors) continue to converge, even when the residual is contaminated by error. The same is true of the shift-invert variant of Arnoldi's method. In this paper we give a mathematical analysis of these new methods. dc.format.extent 318799 bytes dc.format.mimetype application/pdf dc.language.iso en_US en dc.relation.ispartofseries UM Computer Science Department en dc.relation.ispartofseries CS-TR-4890 en dc.relation.ispartofseries UMIACS en dc.relation.ispartofseries UMIACS-TR-2007-45 en dc.title Analysis of the Residual Arnoldi Method en dc.type Technical Report en
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