Analysis of eigenvalue condition numbers for a class of randomized numerical methods for singular matrix pencils

Kressner, Daniel (author), Plestenjak, Bor (author)

The numerical solution of the generalized eigenvalue problem for a singular matrix pencil is challenging due to the discontinuity of its eigenvalues. Classically, such problems are addressed by first extracting the regular part through the staircase form and then applying a standard solver, such as the QZ algorithm, to that regular part. Recently, several novel approaches have been proposed to transform the singular pencil into a regular pencil by relatively simple randomized modifications. In this work, we analyze three such methods by Hochstenbach, Mehl, and Plestenjak that modify, project, or augment the pencil using random matrices. All three methods rely on the normal rank and do not alter the finite eigenvalues of the original pencil. We show that the eigenvalue condition numbers of the transformed pencils are unlikely to be much larger than the $\delta$-weak eigenvalue condition numbers, introduced by Lotz and Noferini, of the original pencil. This not only indicates favorable numerical stability but also reconfirms that these condition numbers are a reliable criterion for detecting simple finite eigenvalues. We also provide evidence that, from a numerical stability perspective, the use of complex instead of real random matrices is preferable even for real singular matrix pencils and real eigenvalues. As a side result, we provide sharp left tail bounds for a product of two independent random variables distributed with the generalized beta distribution of the first kind or Kumaraswamy distribution.

• Language: English
• Keywords: singular pencil, singular generalized eigenvalue problem, eigenvalue condition number, randomized numerical method, random matrices
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2024
• Number of pages: 27 str.
• Numbering: Vol. 64, iss. 3, art. 32
• PID: 20.500.12556/RUL-160120
• UDC: 519.6
• ISSN on article: 0006-3835
• DOI: 10.1007/s10543-024-01033-w
• COBISS.SI-ID: 204416003
• Publication date in RUL: 21.08.2024

Record is a part of a journal

• Title: BIT numerical mathematics
• Shortened title: BIT
• Publisher: Springer Nature, BIT Foundation
• ISSN: 0006-3835
• COBISS.SI-ID: 25103872

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: singularni šop, singularni posplošeni problem lastnih vrednosti, pogojenostno število lastne vrednosti, verjetnostna numerična metoda, naključne matrike