An application of Gelfand's formula in approximating the roots of polynomials

TL;DR

This paper demonstrates how Gelfand's formula and balancing techniques can be used to refine bounds on the roots of polynomials by analyzing their companion matrices, with practical examples included.

Contribution

It introduces a novel application of Gelfand's formula combined with balancing to improve spectral bounds of polynomial roots.

Findings

Enhanced bounds on polynomial roots using Gelfand's formula

Application of balancing techniques to companion matrices

Practical examples illustrating the method's effectiveness

Abstract

The purpose of this paper is to show how Gelfand's formula and balancing can be used to improve the upper and lower bounds of the spectrum of a companion matrix associated with a given real or complex polynomial. Examples and other related ideas are provided.