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

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.