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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]
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Database: Engineering Source
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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]
ISSN:00243795
DOI:10.1016/j.laa.2026.01.014