Convergence of Arnoldi’s method for generalized eigenvalue problems

Abstract

By constructing a-posteriori residual bounds, this paper consider the convergence of implicitly restarted Arnoldi’s methods for generalized eigenvalue problems. Such bounds have been less studied in comparison to bounds on the angle between an eigenvector and the Krylov subspace. Numerical validations of the bounds are given and both cases of convergence and non-convergence are illustrated for the shift-and-invert Arnoldi method and its refined variant. Alternative stopping criteria are also proposed for the Arnoldi methods.