Local convergence analysis of several inexact Newton-type algorithms for general nonlinear eigenvalue problems

Abstract

We study the local convergence of several inexact numerical algorithms closely related to Newton's method for the solution of a simple eigenpair of the general nonlinear eigenvalue problem $$T(\lambda )v=0$$ . We investigate inverse iteration, Rayleigh quotient iteration, residual inverse iteration, and the single-vector Jacobi---Davidson method, analyzing the impact of the tolerances chosen for the approximate solution of the linear systems arising in these algorithms on the order of the local convergence rates. We show that the inexact algorithms can achieve the same order of convergence as the exact methods if appropriate sequences of tolerances are applied to the inner solves. We discuss the connections and emphasize the differences between the standard inexact Newton's method and these inexact algorithms. When the local symmetry of $$T(\lambda )$$ is present, the use of a nonlinear Rayleigh functional is shown to be fundamental in achieving higher order of convergence rates. The convergence results are illustrated by numerical experiments.