Convergence orders of iterative methods for nonlinear eigenvalue problems

The convergence analysis of iterative methods for nonlinear eigenvalue problems is in the most cases restricted either to algebraic simple eigenvalues or to polynomial eigenvalue problems. In this paper we consider two classical methods for general holomorphic eigenvalue problems, namely the nonlinear generalized Rayleigh quotient iteration (NGRQI) and the augmented Newton method. For both methods we prove local quadratic convergence for semi–simple eigenvalues. For defective eigenvalues local linear convergence is shown for the NGRQI. The key tool of our analysis is the representation of the eigenvalues as poles of the resolvent which is a classical result in operator theory. The convergence orders of the mentioned methods depend on the order of the poles of the resolvent. In numerical experiments the theoretical results are verified.