Constructing strong linearizations of matrix polynomials expressed in the Chebyshev bases

Lawrence, Piers and Perez, Javier (2016) Constructing strong linearizations of matrix polynomials expressed in the Chebyshev bases. [MIMS Preprint]

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Abstract

The need to solve polynomial eigenvalue problems for matrix polynomials expressed in nonmonomial bases has become a very important problem. 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 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 whenever the matrix polynomials are symmetric.

Item Type:	MIMS Preprint
Uncontrolled Keywords:	Chebyshev polynomials, Chebyshev pencils, strong linearizations, matrix polynomials, singular matrix polynomials, eigenvector recovery, minimal bases, minimal indices, one-sided factorizations, structure-preserving linearizations
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User:	Javier Perez Alvaro
Date Deposited:	01 Mar 2016
Last Modified:	08 Nov 2017 18:18
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/2443

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