On general convergence in extracting radicals via a fundamental family of iteration functions

Newton's method is well-known to be generally convergent for solving xⁿ - c = 0. In this paper, we first extend this result to the next two members of an infinite family of high order methods referred to here as the Basic Family which starts with Newton's method. While computing roots of unity numerically is a trivial task, studying the general convergence of the Basic Family in this simple case is an important first step toward the understanding of the global behavior of this fundamental family. With the aid of polynomiography, techniques for the visualization of polynomial root-finding, we further conjecture the general convergence of all members of the Basic Family when extracting radicals. Using the computer algebra system Maple, we obtain some partial results toward the proof of our conjecture.