Two-sided and alternating Jacobi-Davidson

Abstract

We discuss two variants of a two-sided Jacobi-Davidson method, which have asymptotically cubic convergence for nonnormal matrices and aim to find both right and left eigenvectors. These methods can be seen as Jacobi-Davidson analogs of Ostrowski’s two-sided Rayleigh Quotient Iteration. Some relations between (exact and inexact) two-sided Jacobi-Davidson and (exact and inexact) two-sided Rayleigh Quotient Iteration are given, together with convergence rates. Furthermore, we introduce an alternating Jacobi-Davidson process, that can be seen as the Jacobi-Davidson analog of Parlett’s alternating Rayleigh Quotient Iteration. The methods are extended to the generalized and polynomial eigenproblem. Advantages of the methods are illustrated by numerical examples.