Generalized Computation of Schröder Iteration Functions To Motivate Families of Julia and Mandelbrot-Like Sets

Abstract

Schroder iteration functions, a generalization of the Newton--Raphson method to determine roots of equations, are generally rational functions which possess some critical points free to converge to attracting cycles. These free critical points, however, satisfy some higher-degree polynomial equations. We present a new algorithmic construction to compute in general all of the Schroder functions' terms as well as to maximize the computational efficiency of these functions associated with a one-parameter family of cubic polynomials. Finally, we examine the Julia sets of the Schroder functions constructed to converge to the nth roots of unity, these roots' basins of attraction, and the orbits of all free critical points of these functions for order higher than four, as applied to the one-parameter family of cubic polynomials mentioned above.