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Numerical Algorithms

, Volume 79, Issue 1, pp 311–335 | Cite as

Disguised and new quasi-Newton methods for nonlinear eigenvalue problems

  • E. Jarlebring
  • A. Koskela
  • G. Mele
Open Access
Original Paper
  • 184 Downloads

Abstract

In this paper, we take a quasi-Newton approach to nonlinear eigenvalue problems (NEPs) of the type M(λ)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, we 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.

Keywords

Nonlinear eigenvalue problems Inverse iteration Iterative methods Quasi-Newton methods 

Notes

Acknowledgements

We thank Wim Michiels (KU Leuven) for valuable discussions regarding partial fraction expansions for time-delay systems.

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.KTH Royal Institute of TechnologyStockholmSweden

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