Stopping Criteria for Eigensolvers

Abstract

In most iterative methods for solving linear systems, the stopping criterion is based on the backward error. In this paper, after having recalled the situation for linear systems and showed that backward analysis can be used on eigenproblems, we present similar stopping criteria for eigensolvers. We also present link between the backward error and the forward error using the condition number. Numerical experiments with symmetric eigensolvers (Jacobi and Lanczos method) and with nonsymmetric ones (QR algorithm, subspace iteration and Arnoldi-Tchebycheff method) illustrate the choice of the backward error. 1 Introduction Physical problems that scientists want to model on a computer often lead to a numerical problem which corresponds to either : 1. solving a linear system, i.e., given a matrix A and a vector b, find x such that Ax = b, or 2. solving an eigenproblem, i.e., given a matrix A, find the eigenpair (; x) such that Ax = x and x 6= 0. For the first class of problems, one can use..