Approximation of pseudospectra of block triangular matrices

Roy, Nandita; Karow, Michael; Bora, Shreemayee; Armentia, Gorka

doi:10.1016/j.laa.2020.08.008

Linear Algebra and its Applications

2021

DOI: 10.1016/j.laa.2020.08.008

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Approximation of pseudospectra of block triangular matrices

Nandita Roy

Michael Karow

Shreemayee Bora

et al.

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2023

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“…Following in the spirit of using sep λ (A, B) in the analysis of the stability of invariant subspaces of matrices [17,18,46], Karow and Kressner used sep D λ (A, B) in 2014 as a tool in deriving improved perturbation bounds [27]. Most recently in 2021, Roy et al [40] used sep V λ (A, B) in connection with approximating pseudospectra of block triangular matrices; in this case, the value of sep V λ (A, B) can be used to construct several different outer approximations to pseudospectra of these structured matrices.…”

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Fast computation of Sep$_\lambda$ via interpolation-based globality certificates

Mitchell¹

2023

etna

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Given two square matrices A and B, we propose a new approach for computing the smallest value ε ≥ 0 such that A + E and A + F share an eigenvalue, where E = F = ε. In 2006, Gu and Overton proposed the first algorithm for computing this quantity, called sep λ (A, B) ("sep-lambda"), using ideas inspired from an earlier algorithm of Gu for computing the distance to uncontrollability. However, the algorithm of Gu and Overton is extremely expensive, which limits it to the tiniest of problems, and until now, no other algorithms have been known. Our new algorithm can be orders of magnitude faster and can solve problems where A and B are of moderate size. Moreover, our method consists of many "embarrassingly parallel" computations, and so it can be further accelerated on multi-core hardware. Finally, we also propose the first algorithm to compute an earlier version of sep-lambda where E + F = ε.

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confidence: 99%

Fast computation of Sep$_\lambda$ via interpolation-based globality certificates

Mitchell¹

2023

etna

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Approximation of pseudospectra of block triangular matrices

Cited by 1 publication

References 9 publications

Fast computation of Sep$_\lambda$ via interpolation-based globality certificates

Fast computation of Sep$_\lambda$ via interpolation-based globality certificates

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