Previous analyses of Laguerre’s iteration method have provided results on the behavior of this popular method when applied to the polynomials 𝑝𝑛(𝑧) = 𝑧𝑛 − 1, 𝑛 ∈ N. In this paper, we summarize known analytical results and provide new results. In particular, we study symmetry properties of the Laguerre iteration function and clarify the dynamics of the method. We show analytically and demonstrate computationally that for each 𝑛 ≥ 5 the basin of attraction to the roots is a subset of an annulus that contains the unit circle and whose Lebesgue measure shrinks to zero as 𝑛 → ∞. We obtain a good estimate of the size of the bounding annulus. We show that the boundary of the basin of convergence exhibits fractal nature and quasi self-similarity. We also discuss the connectedness of the basin for large values of 𝑛. We also numerically nd some short nite cycles on the boundary of the basin of convergence for 𝑛 = 5, ..., 8. Finally, we demonstrate that when using the oating point arithmetic and the general formulation of the method, convergence occurs even from starting values outside of the basin of convergence due to the loss of signi cance during the evaluation of the iteration function.
Bělík, Pavel; Kang, HeeChan; Walsh, Andrew; and Winegar, Emma, "On the Dynamics of Laguerre’s Iteration Method for Finding the nth Roots of Unity" (2014). Faculty Authored Articles. 1.
This article was originally published in the International Journal of Computational Mathematics at http://dx.doi.org/10.1155/2014/321585