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Mathematics > Numerical Analysis

arXiv:2208.02057 (math)
[Submitted on 3 Aug 2022 (v1), last revised 29 Aug 2023 (this version, v2)]

Title:A new deflation criterion for the QZ algorithm

Authors:Thijs Steel, Raf Vandebril, Julien Langou
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Abstract:The QZ algorithm computes the Schur form of a matrix pencil. It is an iterative algorithm and at some point, it must decide that an eigenvalue has converged and move on with another one. Choosing a criterion that makes this decision is nontrivial. If it is too strict, the algorithm might waste iterations on already converged eigenvalues. If it is not strict enough, the computed eigenvalues might be inaccurate. Additionally, the criterion should not be computationally expensive to evaluate. This paper introduces a new criterion based on the size of and the gap between the eigenvalues. This is similar to the work of Ahues and Tissuer for the QR algorithm. Theoretical arguments and numerical experiments suggest that it outperforms the most popular criteria in terms of accuracy. Additionally, this paper evaluates some commonly used criteria for infinite eigenvalues.
Comments: 11 pages, 6 figures
Subjects: Numerical Analysis (math.NA)
MSC classes: 65F15
Cite as: arXiv:2208.02057 [math.NA]
  (or arXiv:2208.02057v2 [math.NA] for this version)
  https://doi.org/10.48550/arXiv.2208.02057
Journal reference: Numer Linear Algebra Appl. 2023;e2524
Related DOI: https://doi.org/10.1002/nla.2524

Submission history

From: Thijs Steel [view email]
[v1] Wed, 3 Aug 2022 13:26:00 UTC (12 KB)
[v2] Tue, 29 Aug 2023 10:13:56 UTC (139 KB)
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