Extraneous fixed points, basin boundaries and chaotic dynamics for Schr�der and K�nig rational iteration functions

Abstract

Summary. The Schr6der and K6nig iteration schemes to find the zeros of a (polynomial) function g(z) represent generalizations of Newton's method. In both schemes, iteration functions fro(Z) are constructed so that sequences z,+~=f,,(z,) converge locally to a root z* of g(z) as O(Iz,--z*f"). It is well known that attractive cycles, other than the zeros z*, may exist for Newton's method (m = 2). As m increases, the iteration functions add extraneous fixed points and cycles. Whether attractive or repulsive, they affect the Julia set basin boundaries. The K6nig functions K,,(z) appear to minimize such perturbations. In the case of two roots, e.g. g(z)= z 2-1, Cayley's classical result for the basins of attraction of Newton's method is extended for all Kin(z).The existence of chaotic {z,} sequences is also demonstrated for these iteration methods.

Basins of attraction for Zhou–Chen–Song fourth order family of methods for multiple roots

“…Many iterative methods have fixed points that are not zeros of the function of interest. Those points are called extraneous fixed points (see Vrscay and Gilbert [26]). Those points could be attractive which will trap an iteration sequence and give erroneous results.…”

“…Vrscay and Gilbert [26] show that if the points are attractive then the method will give erroneous results. If the points are repulsive then the method may not converge to a root near the initial guess.…”

A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points

“…Even in the case of repulsive or indifferent fixed points, an initial value x 0 chosen near a desired root may converge to another unwanted remote root. Indeed, these aspects of the Schröder functions [43] were observed in an application to the same family of…”

“…} by Vrscay and Gilbert [43] . Especially the presence of attractive cycles induced by the extraneous fixed points of R f may alter the basins of attraction due to the trapped sequence { x n }.…”

“…In this section, we will investigate the extraneous fixed points [22,43] of the iterative map (3.1) and relevant dynamics associated with their basins of attraction. The dynamics underlying basins of attraction was initiated by Stewart [41] and followed by works of Amat et al [2][3][4][5] , Scott et al [38] , Chun et al [10] , Chun-Neta [11] , Chicharro et al [8] , Cordero et al [15] , Neta et al [28,33] , Argyros-Magreñan [7] , Magreñan [27] , Magreñan et al [26] , Andreu et al [6] and Chun et al [12] .…”