TY - JOUR

T1 - Constructing strong linearizations of matrix polynomials expressed in chebyshev bases

AU - Lawrence, Piers W.

AU - Pérez, Javier

N1 - Funding Information:
∗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 (piers.lawrence@cs.kuleuven.be). ‡Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium (javier.perezalvaro@kuleuven.be).
Publisher Copyright:
© 2017 Society for Industrial and Applied Mathematics.

PY - 2017

Y1 - 2017

N2 - 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).

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).

KW - Chebyshev pencils

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

UR - http://www.scopus.com/inward/record.url?scp=85031827031&partnerID=8YFLogxK

U2 - 10.1137/16M105839X

DO - 10.1137/16M105839X

M3 - Article

AN - SCOPUS:85031827031

SN - 0895-4798

VL - 38

SP - 683

EP - 709

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 3

ER -