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Convergence factors of Newton methods for nonlinear eigenvalue problems. (English) Zbl 1250.65072
Author’s abstract: Consider a complex sequence \(\{\lambda_k\}_{k=0}^{\infty}\) convergent to \(\lambda _{\ast }\in \mathbb{C}\) with order \(p\in N\). The convergence factor is typically defined as the fraction \(c_{k}:=(\lambda _{k+1}-\lambda _{\ast })/(\lambda _{k}-\lambda _{\ast })^{p}\) in the limit \(k\to \infty \). In this paper, we prove formulas characterizing \(c_{k}\) in the limit \(k\to \infty \) 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 \(c_{k}\) 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 \(c_{k}\) as k\(\to \infty \) does not exist. For cases where this limit does not appear to exist, we prove other limiting expressions such that a characterization of \(c_{k}\) in the limit is still possible.

65H17 Numerical solution of nonlinear eigenvalue and eigenvector problems
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