Vol. 1, Issue 3, Part C (2015)

Abstract
Polynomial root finding is the origin of some fundamental discoveries in Mathematics and sciences. Where analytical or exact method fails, numerical approximation methods often succeed. We will investigate the numerical root finding method of Newton from a dynamical system perspective. Consider a complex valued function f and pick x₀ in the domain of f. If we iterate this function around x₀, then we have sequence x₀, f (x₀), f ² (x₀), . . . which becomes a dynamical system. We are essentially interested in the long term behavior of this system. We address the connection between behavior of both Newton’s and Halley’s methods with Voronoi diagram of underlying polynomial roots.