Constructing strong linearizations of matrix polynomials expressed in chebyshev bases

Piers W. Lawrence; Javier Pérez

doi:10.1137/16M105839X

Constructing strong linearizations of matrix polynomials expressed in chebyshev bases

Piers W. Lawrence
, Javier Pérez

Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

The need to solve polynomial eigenvalue problems for matrix polynomials expressed in nonmonomial bases has become very important. Among the most important bases in numerical applications are the Chebyshev polynomials of the first and second kind. In this work, we introduce a new approach for constructing strong linearizations for matrix polynomials expressed in Chebyshev bases, generalizing the classical colleague pencil, and expanding the arena in which to look for linearizations of matrix polynomials expressed in Chebyshev bases. We show that any of these linearizations is a strong linearization regardless of whether the matrix polynomial is regular or singular. In addition, we show how to recover eigenvectors, minimal indices, and minimal bases of the polynomial from those of any of the new linearizations. As an example, we also construct strong linearizations for matrix polynomials of odd degree that are symmetric (resp., Hermitian) whenever the matrix polynomials are symmetric (resp., Hermitian).

Original language	English
Pages (from-to)	683-709
Number of pages	27
Journal	SIAM Journal on Matrix Analysis and Applications
Volume	38
Issue number	3
DOIs	https://doi.org/10.1137/16M105839X
State	Published - 2017

Funding

∗Received by the editors January 25, 2016; accepted for publication (in revised form) by B. Meini April 3, 2017; published electronically July 27, 2017. http://www.siam.org/journals/simax/38-3/M105839.html Funding: The work of the first author was partially supported by the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office. The work of the second author was supported by Engineering and Physical Sciences Research Council grant EP/I005293. †Department of Mathematical Engineering, Universitécatholique de Louvain, B-1348 Louvain-la-Neuve, Belgium, and Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium ([email protected]). ‡Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium ([email protected]).

Funders	Funder number
Engineering and Physical Sciences Research Council	EP/I005293

Keywords

Chebyshev pencils
Chebyshev polynomials
Eigenvector recovery
Matrix polynomials
Minimal bases
Minimal indices
One-sided factorizations
Singular matrix polynomials
Strong linearizations
Structure-preserving linearizations

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10.1137/16M105839X

Cite this

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title = "Constructing strong linearizations of matrix polynomials expressed in chebyshev bases",

abstract = "The need to solve polynomial eigenvalue problems for matrix polynomials expressed in nonmonomial bases has become very important. Among the most important bases in numerical applications are the Chebyshev polynomials of the first and second kind. In this work, we introduce a new approach for constructing strong linearizations for matrix polynomials expressed in Chebyshev bases, generalizing the classical colleague pencil, and expanding the arena in which to look for linearizations of matrix polynomials expressed in Chebyshev bases. We show that any of these linearizations is a strong linearization regardless of whether the matrix polynomial is regular or singular. In addition, we show how to recover eigenvectors, minimal indices, and minimal bases of the polynomial from those of any of the new linearizations. As an example, we also construct strong linearizations for matrix polynomials of odd degree that are symmetric (resp., Hermitian) whenever the matrix polynomials are symmetric (resp., Hermitian).",

keywords = "Chebyshev pencils, Chebyshev polynomials, Eigenvector recovery, Matrix polynomials, Minimal bases, Minimal indices, One-sided factorizations, Singular matrix polynomials, Strong linearizations, Structure-preserving linearizations",

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AB - The need to solve polynomial eigenvalue problems for matrix polynomials expressed in nonmonomial bases has become very important. Among the most important bases in numerical applications are the Chebyshev polynomials of the first and second kind. In this work, we introduce a new approach for constructing strong linearizations for matrix polynomials expressed in Chebyshev bases, generalizing the classical colleague pencil, and expanding the arena in which to look for linearizations of matrix polynomials expressed in Chebyshev bases. We show that any of these linearizations is a strong linearization regardless of whether the matrix polynomial is regular or singular. In addition, we show how to recover eigenvectors, minimal indices, and minimal bases of the polynomial from those of any of the new linearizations. As an example, we also construct strong linearizations for matrix polynomials of odd degree that are symmetric (resp., Hermitian) whenever the matrix polynomials are symmetric (resp., Hermitian).

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KW - Chebyshev polynomials

KW - Eigenvector recovery

KW - Matrix polynomials

KW - Minimal bases

KW - Minimal indices

KW - One-sided factorizations

KW - Singular matrix polynomials

KW - Strong linearizations

KW - Structure-preserving linearizations

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Constructing strong linearizations of matrix polynomials expressed in chebyshev bases

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