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

Abstract

Schröder 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 Schröder 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 Schröder 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. Introduction.A number of mathematical applications lead to the problem of finding the roots of some equation using iterative methods. Unfortunately, the convergence of an iterative method is not assured independently of the starting value. There exist certain starting values that are not suitable in yielding the desired result. Looking at the set of all the starting values from a geometrical point of view will definitely help us to choose these initial approximations to a root and compare the convergence properties of various iterative methods.Ernst Schröder [12] in 1870 (see also [7]) described a method of finding a rational iterating function of any desired "order of convergence" to determine roots of equations. For polynomial equations this involves the iteration of rational functions over the Riemann sphere which is described by the classical theory of Julia [9] and Fatou [6] and its subsequent developments, also of paramount importance in the context of numerical analysis. In what follows we abbreviate as f k the k-fold composition f • f • · · · • f , by region we mean a connected open set on the extended complex plane C = C ∪ {∞}, and if we have a rational function of the form R = P/Q, where P and Q are complex polynomials with no common factors, the degree of R is defined by deg(R) = max{deg(P ), deg(Q)}.It appears that, for cases of practical interest, convergence of the sequence of iterates

Dynamics of the King and Jarratt iterations

“…. ), associated to the one-parameter family p A , some progress in the study of the A-parameter space has been achieved in [6], [7], [17], and [18].…”

Purely Iterative Algorithms for Newton’s Maps and General Convergence

“…The following family of iterative functions represents Newton's method, Chebyshev's iterative function, Halley's method, Super-Halley, c-iterative function (considering θ = 0 below) and Chebyshev-Halley family, among others. See for instance [22,[29][30][31][32][33][34][35][36][37][38]. The family of iterative methods given by…”

Δυναμική Ρητών Επαναληπτικών Μεθόδων Και Μορφοκλασματικές Συναρτήσεις

“…Οι επαναληπτικές συναρτήσεις Schroder αποτελούν μία οικογένεια ρητών επαναληπτικών συναρτή σεων, οι οποίες είναι σχεδιασμένες, ώστε να συγκλίνουν με τάξη σ προς τις ρίζες μιας συνάρτησης / για κάθε σ > 2. Μέρος αυτού του εδαφίου έχει αναπτυχθεί στην [59] και μπορεί να θεωρηθεί επέκταση της [156] για συναρτήσεις Schroder τάξης υψηλότερης από τέσσερα.…”