Constructing strong linearizations of matrix polynomials expressed in chebyshev bases

Piers W. Lawrence, Javier Pérez

Research output: Contribution to journalArticlepeer-review

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 languageEnglish
Pages (from-to)683-709
Number of pages27
JournalSIAM Journal on Matrix Analysis and Applications
Volume38
Issue number3
DOIs
StatePublished - 2017

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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