Analysis of the Residual Arnoldi Method

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.