Condition Numbers of Hessenberg Companion Matrices

TL;DR

This paper analyzes the condition numbers of Hessenberg companion matrices, especially Fiedler matrices, and introduces methods to compare and improve their numerical stability.

Contribution

It presents a straightforward approach to compare condition numbers of Fiedler matrices and identifies other companion matrices with potentially smaller condition numbers.

Findings

Hessenberg form simplifies condition number comparison

Some companion matrices have smaller condition numbers than Fiedler matrices

Perturbed Frobenius matrices can maintain polynomial roots with better conditioning

Abstract

The Fiedler matrices are a large class of companion matrices that include the well-known Frobenius companion matrix. The Fiedler matrices are part of a larger class of companion matrices that can be characterized with a Hessenberg form. In this paper, we demonstrate that the Hessenberg form of the Fiedler companion matrices provides a straight-forward way to compare the condition numbers of these matrices. We also show that there are other companion matrices which can provide a much smaller condition number than any Fiedler companion matrix. We finish by exploring the condition number of a class of matrices obtained from perturbing a Frobenius companion matrix while preserving the characteristic polynomial.