Bounds on polynomial roots using intercyclic companion matrices

TL;DR

This paper investigates new bounds on polynomial roots derived from intercyclic companion matrices, improving existing bounds in certain cases and providing methods to determine optimal bounds within specific matrix classes.

Contribution

It introduces novel bounds from intercyclic companion matrices, compares their effectiveness to Frobenius matrices, and explores bounds from polynomial transformations and inverses.

Findings

Intercyclic companion matrices can improve root bounds by up to a factor of two over Frobenius matrices.

Hessenberg form helps identify optimal Fiedler matrix bounds using the infinity norm.

Considering polynomial powers and inverses can yield significantly better bounds for certain polynomials.

Abstract

The Frobenius companion matrix, and more recently the Fiedler companion matrices, have been used to provide lower and upper bounds on the modulus of any root of a polynomial $p(x)$. In this paper we explore new bounds obtained from taking the $1$-norm and $\infty$-norm of a matrix in the wider class of intercyclic companion matrices. As is the case with Fiedler matrices, we observe that the new bounds from intercyclic companion matrices can improve those from the Frobenius matrix by at most a factor of two. By using the Hessenberg form of an intercyclic companion matrix, we describe how to determine the best upper bound when restricted to Fiedler companion matrices using the $\infty$-norm. We also obtain a new general bound by considering the polynomial $x^qp(x)$ for $q>0$. We end by considering upper bounds obtained from inverses of monic reversal polynomials of intercyclic companion matrices, noting that these can make more significant improvements on the bounds from a Frobenius companion matrix for certain polynomials.