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Disguised and new quasi-Newton methods for nonlinear eigenvalue problems

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.

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Acknowledgements

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

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Correspondence to E. Jarlebring.

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Jarlebring, E., Koskela, A. & Mele, G. Disguised and new quasi-Newton methods for nonlinear eigenvalue problems. Numer Algor 79, 311–335 (2018). https://doi.org/10.1007/s11075-017-0438-2

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Keywords

  • Nonlinear eigenvalue problems
  • Inverse iteration
  • Iterative methods
  • Quasi-Newton methods