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Title:Analysis of eigenvalue condition numbers for a class of randomized numerical methods for singular matrix pencils
Authors:ID Kressner, Daniel (Author)
ID Plestenjak, Bor (Author)
Files:.pdf PDF - Presentation file, download (659,18 KB)
MD5: AD9B162A12460C210119CB3B30EC249B
 
URL URL - Source URL, visit https://link.springer.com/article/10.1007/s10543-024-01033-w
 
Language:English
Typology:1.01 - Original Scientific Article
Organization:Logo IMFM - Institute of Mathematics, Physics, and Mechanics
Abstract: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 δ-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.
Keywords:singular pencil, singular generalized eigenvalue problem, eigenvalue condition number, randomized numerical method, random matrices
Publication status:Published
Publication version:Version of Record
Publication date:01.09.2024
Year of publishing:2024
Number of pages:27 str.
Numbering:Vol. 64, iss. 3, article no. 32
PID:20.500.12556/DiRROS-20224 New window
UDC:519.6
ISSN on article:0006-3835
DOI:10.1007/s10543-024-01033-w New window
COBISS.SI-ID:204416003 New window
Note:
Publication date in DiRROS:26.08.2024
Views:389
Downloads:185
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KRESSNER, Daniel and PLESTENJAK, Bor, 2024, Analysis of eigenvalue condition numbers for a class of randomized numerical methods for singular matrix pencils. BIT Numerical Mathematics [online]. 2024. Vol. 64, no. 3,  32. [Accessed 6 April 2025]. DOI 10.1007/s10543-024-01033-w. Retrieved from: https://dirros.openscience.si/IzpisGradiva.php?lang=eng&id=20224
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Record is a part of a journal

Title:BIT Numerical Mathematics
Shortened title:BIT
Publisher:Springer
ISSN:0006-3835
COBISS.SI-ID:25103872 New window

Document is financed by a project

Funder:SNSF - Swiss National Science Foundation
Funding programme:Projects
Project number:192049
Name:Probabilistic methods for joint and singular eigenvalue problems
Funder:ARIS - Slovenian Research and Innovation Agency
Project number:N1-0154
Name:Verjetnostne metode za skupne in singularne probleme lastnih vrednosti
Funder:ARIS - Slovenian Research and Innovation Agency
Project number:P1-0294
Name:Računsko intenzivne metode v teoretičnem računalništvu, diskretni matematiki, kombinatorični optimizaciji ter numerični analizi in algebri z uporabo v naravoslovju in družboslovju

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


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