A Jacobi–Davidson like method for nonlinear eigenvalue problems based on singularity theory

Abstract

We present a Jacobi-Davidson like correction formula for left and right eigenvector approximations for non-Hermitian nonlinear eigenvalue problems. It exploits techniques from singularity theory for characterizing singular points of nonlinear equations. Unlike standard nonlinear Jacobi-Davidson, the correction formula does not contain derivative information and works with orthogonal projectors only. Moreover, the basic method is modified in that the new eigenvalue approximation is taken as a nonlinear Rayleigh functional obtained as root of a certain scalar nonlinear equation the existence of which-as well as a first order perturbation expansion-is shown.