# Analysis of the Residual Arnoldi Method

 Title Analysis of the Residual Arnoldi Method Publication Type Reports Year of Publication 2007 Authors Lee C-R, Stewart G.W Date Published 2007/10/15/ Institution Instititue for Advanced Computer Studies, Univ of Maryland, College Park Keywords Technical Report Abstract The Arnoldi method generates a nested squences of orthonormal bases$U_{1},U_{2}, \ldots$ 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. URL http://drum.lib.umd.edu/handle/1903/7428