Disguised and new Quasi-Newton methods for nonlinear eigenvalue problems

Elias Jarlebring; Antti Koskela; Giampaolo Mele

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arXiv · 1702.08492

Disguised and new Quasi-Newton methods for nonlinear eigenvalue problems

Elias Jarlebring·Antti Koskela·Giampaolo Mele

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Abstract

In this paper we take a quasi-Newton approach to nonlinear eigenvalue problems (NEPs) of the type $M(\lambda)v=0$, where $M:\mathbb{C}\rightarrow\mathbb{C}^{n\times n}$ is a holomorphic function. We investigate which types of approximations of the Jacobian matrix lead to competitive algorithms, and provide convergence theory. The convergence analysis is based on theory for quasi-Newton methods and Keldysh's theorem for NEPs. We derive new algorithms and also show that several well-established methods for NEPs can be interpreted as quasi-Newton methods, and thereby provide insight to their convergence behavior. In particular, we establish quasi-Newton interpretations of Neumaier's residual inverse iteration and Ruhe's method of successive linear problems.

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Elias Jarlebring, Antti Koskela, Giampaolo Mele. 2017-02-27. Disguised and new Quasi-Newton methods for nonlinear eigenvalue problems. https://arxiv.org/abs/1702.08492

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