A rational-Chebyshev projection method for nonlinear eigenvalue problems

Ziyuan Tang, Yousef Saad

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

This article describes a projection method based on a combination of rational and polynomial approximations for efficiently solving large nonlinear eigenvalue problems. In a first stage the nonlinear matrix function (Formula presented.) under consideration is approximated by a matrix polynomial in (Formula presented.). The error resulting from this polynomial approximation is in turn approximated by rational functions with the help of the Cauchy integral formula. The two approximations are combined and a linearization is performed. A key ingredient of the proposed approach is a projection method that uses subspaces spanned by vectors of the same dimension as that of the original problem instead of that of the linearized problem. A procedure is also presented to automatically select shifts and to partition the region of interest into a few subregions. This allows to subdivide the problem into smaller subproblems that are solved independently. The accuracy of the proposed method is theoretically analyzed and its performance is illustrated with a few test problems that have been discussed in the literature.

Original languageEnglish (US)
Article numbere2563
JournalNumerical Linear Algebra with Applications
Volume31
Issue number6
DOIs
StateAccepted/In press - 2024

Bibliographical note

Publisher Copyright:
© 2024 The Authors. Numerical Linear Algebra with Applications published by John Wiley & Sons Ltd.

Keywords

  • Cauchy integral formula
  • Krylov subspace method
  • nonlinear eigenvalue problem
  • polynomial approximation
  • projection methods
  • rational approximation
  • subspace iteration method

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