Fast and inverse-free algorithms for deflating subspaces.
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| Title: | Fast and inverse-free algorithms for deflating subspaces. |
|---|---|
| Authors: | Demmel, James1 (AUTHOR), Dumitriu, Ioana2 (AUTHOR), Schneider, Ryan1,3 (AUTHOR) ryan.schneider@berkeley.edu |
| Source: | Linear Algebra & its Applications. Apr2026, Vol. 735, p222-258. 37p. |
| Subjects: | Matrix pencils, Invariant subspaces, Numerical solutions for linear algebra, Matrix functions, Numerical analysis |
| Abstract: | This paper explores a key question in numerical linear algebra: how can we compute projectors onto the deflating subspaces of a regular matrix pencil (A , B) , in particular without using matrix inversion or defaulting to an expensive Schur decomposition? We focus specifically on spectral projectors, whose associated deflating subspaces correspond to sets of eigenvalues/eigenvectors. In this work, we present a high-level approach to this problem that combines rational function approximation with an inverse-free arithmetic of Benner and Byers [Numerische Mathematik 2006]. The result is a numerical framework that captures existing inverse-free methods, generates an array of new options, and provides straightforward tools for pursuing efficiency on structured problems (e.g., definite pencils). To exhibit the efficacy of this framework, we consider a handful of methods in detail, including Implicit Repeated Squaring and iterations based on the matrix sign function. [ABSTRACT FROM AUTHOR] |
| Copyright of Linear Algebra & its Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 191426587 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Fast and inverse-free algorithms for deflating subspaces. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Demmel%2C+James%22">Demmel, James</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Dumitriu%2C+Ioana%22">Dumitriu, Ioana</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Schneider%2C+Ryan%22">Schneider, Ryan</searchLink><relatesTo>1,3</relatesTo> (AUTHOR)<i> ryan.schneider@berkeley.edu</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Linear+Algebra+%26+its+Applications%22">Linear Algebra & its Applications</searchLink>. Apr2026, Vol. 735, p222-258. 37p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Matrix+pencils%22">Matrix pencils</searchLink><br /><searchLink fieldCode="DE" term="%22Invariant+subspaces%22">Invariant subspaces</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+solutions+for+linear+algebra%22">Numerical solutions for linear algebra</searchLink><br /><searchLink fieldCode="DE" term="%22Matrix+functions%22">Matrix functions</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+analysis%22">Numerical analysis</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: This paper explores a key question in numerical linear algebra: how can we compute projectors onto the deflating subspaces of a regular matrix pencil (A , B) , in particular without using matrix inversion or defaulting to an expensive Schur decomposition? We focus specifically on spectral projectors, whose associated deflating subspaces correspond to sets of eigenvalues/eigenvectors. In this work, we present a high-level approach to this problem that combines rational function approximation with an inverse-free arithmetic of Benner and Byers [Numerische Mathematik 2006]. The result is a numerical framework that captures existing inverse-free methods, generates an array of new options, and provides straightforward tools for pursuing efficiency on structured problems (e.g., definite pencils). To exhibit the efficacy of this framework, we consider a handful of methods in detail, including Implicit Repeated Squaring and iterations based on the matrix sign function. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Linear Algebra & its Applications is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.laa.2026.01.014 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 37 StartPage: 222 Subjects: – SubjectFull: Matrix pencils Type: general – SubjectFull: Invariant subspaces Type: general – SubjectFull: Numerical solutions for linear algebra Type: general – SubjectFull: Matrix functions Type: general – SubjectFull: Numerical analysis Type: general Titles: – TitleFull: Fast and inverse-free algorithms for deflating subspaces. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Demmel, James – PersonEntity: Name: NameFull: Dumitriu, Ioana – PersonEntity: Name: NameFull: Schneider, Ryan IsPartOfRelationships: – BibEntity: Dates: – D: 15 M: 04 Text: Apr2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 00243795 Numbering: – Type: volume Value: 735 Titles: – TitleFull: Linear Algebra & its Applications Type: main |
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