Abstract
Recent studies on the numerical methods for solving large eigenvalue problems have shown the Arnoldi-like methods can tolerate various types of errors during computation. One of them is the inexact Arnoldi method and the other one is the residual Arnoldi method. Both methods allow large errors with opposite allowable error patterns. Classical perturbation theorems that use first order approximation are not suitable in the analysis of those methods. In Chapter 1, we develop a perturbation theorem for eigensystems, which makes no assumption on the error size, and use it to analyze the perturbations of both methods. In Chapter 2, we study the Inexact Structure-Preserving Arnoldi Methods (ISPAM) for solving Hermitian and skew-Hermitian eigenvalue problems, by which the solutions can preserve the desirable numerical properties as those by the exact methods. The difference between ISPAM and IAM is in the approximation extraction stage, where ISPAM uses the structured Rayleigh quotients that preserve the structures of the original matrices. We provide the formulation for their backward errors, which are also Hermitian and skew-Hermitian, and analyze their allowable inexactness based on the residual gap hypothesis. The solutions obtained by ISPAM can be as accurate as those computed by IAM, under the same allowable error condition. In Chapter 3, we present an inexact inverse iteration method to find the minimal eigenvalue and the associated eigenvector of an irreducible $M$-matrix. We propose two different relaxation strategies for solving the linear system of inner iterations. For the convergence of these two iterations, we show they are globally linear and superlinear, respectively.