Stability analysis of the two-level orthogonal Arnoldi procedure

Ding Lu, Yangfeng Su, Zhaojun Bai

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42 Scopus citations

Abstract

The second-order Arnoldi (SOAR) procedure is an algorithm for computing an orthonormal basis of the second-order Krylov subspace. It has found applications in solving quadratic eigenvalue problems and model order reduction of second-order dynamical systems among others. Unfortunately, the SOAR procedure can be numerically unstable. The two-level orthogonal Arnoldi (TOAR) procedure has been proposed as an alternative to SOAR to cure the numerical instability. In this paper, we provide a rigorous stability analysis of the TOAR procedure. We prove that under mild assumptions, the TOAR procedure is backward stable in computing an orthonormal basis of the associated linear Krylov subspace. The benefit of the backward stability of TOAR is demonstrated by its high accuracy in structure-preserving model order reduction of second-order dynamical systems.

Original languageEnglish
Pages (from-to)195-214
Number of pages20
JournalSIAM Journal on Matrix Analysis and Applications
Volume37
Issue number1
DOIs
StatePublished - 2016

Bibliographical note

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Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

Keywords

  • Backward stability
  • Dynamical systems
  • Model order reduction
  • Second-order Arnoldi procedure
  • Second-order Krylov subspace

ASJC Scopus subject areas

  • Analysis

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