# Chebyshev-Fiedler pencils

Noferini, Vanni and Perez, Javier (2015) Chebyshev-Fiedler pencils. [MIMS Preprint]

## Abstract

Fiedler pencils are a family of strong linearizations for polynomials expressed in the monomial basis, that include the classical Frobenius companion pencils as special cases. We generalize the definition of a Fiedler pencil from monomials to a larger class of orthogonal polynomial bases. In particular, we derive comrade-Fiedler pencils for two bases that are extremely important in practical applications: the Chebyshev polynomials of the first and second kind. The new approach allows one to construct linearizations having limited bandwidth: a Chebyshev analogue of the pentadiagonal Fiedler pencils in the monomial basis. Moreover, our theory allows for linearizations of square matrix polynomials expressed in the Chebyshev basis (and in other bases), regardless of whether the matrix polynomial is regular or singular, and for recovery formulae for eigenvectors, and minimal indices and bases.

Item Type: MIMS Preprint Fiedler pencil, Chebyshev polynomial, linearization, matrix polynomial, singular matrix polynomial, eigenvector recovery, minimal basis, minimal indices MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theoryMSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis Javier Perez Alvaro 24 Sep 2015 08 Nov 2017 18:18 http://eprints.maths.manchester.ac.uk/id/eprint/2386