Analysis of Laguerre's method applied to find the roots of unity

TL;DR

This paper offers new analytical, computational, and graphical insights into Laguerre's method for finding roots of unity, extending previous convergence analyses and highlighting open questions.

Contribution

It provides additional detailed results and raises new questions about Laguerre's method's behavior on roots of unity beyond prior analyses.

Findings

Enhanced understanding of convergence properties

New computational and graphical insights

Identification of open questions in the method's analysis

Abstract

Previous analyses of Laguerre's method have provided results on the convergence and properties of this popular method when applied to the polynomials $p_n(z)=z^n-1$, $n\in\mathbb{N}$. While these analyses appear to provide a fairly complete picture, careful study of the results reveals that more can be said. We provide additional analytical, computational, and graphical results, details, and insights. We raise and summarize questions that still need to be answered.