Convergence factors of Newton methods for nonlinear eigenvalue problems

Consider a complex sequence {λk}k=0 convergent to λ∗ ∈ C with order p ∈ N. The convergence factor is typically defined as the fraction ck := (λk+1 − λ∗)/(λk − λ∗) in the limit k → ∞. In this paper we prove formulas characterizing ck in the limit k → ∞ for two different Newton-type methods for nonlinear eigenvalue problems. The formulas are expressed in terms of the left and right eigenvectors. The two treated methods are called the method of successive linear problems (MSLP) and augmented Newton and are widely used in the literature. We prove several explicit formulas for ck for both methods. Formulas for both methods are found for simple as well as double eigenvalues. In some cases, we observe in examples that the limit ck as k →∞ does not exist. For cases where this limit appears to not exist, we prove other limiting expressions such that a characterization of ck in the limit is still possible.