Accuracy assessment for eigencomputations: Variety of backward errors and pseudospectra Original Research Article

We show that backward errors and pseudospectra combined together are useful tools to assess the validity of a computed eigenvalue. 1. Given a set τ of admissible perturbations ΔA on a matrix A and a norm on τ (relative or absolute), the backward error η(z) for z as a candidate eigenvalue of A is the smallest size of perturbation ΔA such that z is an exact eigenvalue of A+ΔA. 2. The pseudospectrum associated with a backward error of level ε is image . It contains all the points z which are seen as eigenvalues within an accuracy tolerance of ε. In this paper, we focus on normwise and homotopic perturbations which yield respectively for the approximate eigenvalue μ the backward errors η(μ)=1/short parallelAshort parallelshort parallel(A−μI)−1short parallel and short parallelEshort parallel/short parallelAshort parallelρ(E(A−μI)−1). An application to the Arnoldi method is presented.