| Title: | Randomized methods for computing joint eigenvalues, with applications to multiparameter eigenvalue problems and root finding |
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| Authors: | ID He, Haoze (Author)
ID Kressner, Daniel (Author)
ID Plestenjak, Bor (Author) |
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| Files: |  PDF - Presentation file, download (3,88 MB)
MD5: E66AC2A60ACC80A7E41729BFCCAC7F73
 URL - Source URL, visit https://link.springer.com/article/10.1007/s11075-024-01971-0
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | It is well known that a family of $n \times n$ commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the $n$ joint eigenvalues of the family. In this work, we consider the task of numerically computing approximations to such joint eigenvalues for a family of (nearly) commuting matrices. This task arises, for example, in solvers for multiparameter eigenvalue problems and systems of multivariate polynomials, which are our main motivations. We propose and analyze a simple approach that computes eigenvalues as one-sided or two-sided Rayleigh quotients from eigenvectors of a random linear combination of the matrices in the family. We provide some analysis and numerous numerical examples, showing that such randomized approaches can compute semisimple joint eigenvalues accurately and lead to improved performance of existing solvers. |
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| Keywords: | commuting matrices, joint eigenvalues, Rayleigh quotient, randomized numerical linear algebra, multiparameter eigenvalue problems, polynomial systems |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.11.2025 |
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| Year of publishing: | 2025 |
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| Number of pages: | str. 861–892 |
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| Numbering: | Vol. 100, iss. 3 |
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| PID: | 20.500.12556/DiRROS-23878  |
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| UDC: | 519.6 |
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| ISSN on article: | 1017-1398 |
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| DOI: | 10.1007/s11075-024-01971-0 |
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| COBISS.SI-ID: | 215046403 |
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| Note: |
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| Publication date in DiRROS: | 16.10.2025 |
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| Views: | 212 |
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| Downloads: | 113 |
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| Metadata: |    |
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