On Extraneous Fixed-Points of the Basic Family of Iteration Functions

Abstract

Let K denote either the reals or the complex numbers. Consider the root-finding problem for an analytic function f from K into itself via an iteration function F. An extraneous fixed-point of F is a fixed-point different than a root of f. We prove that all extraneous fixed-points of any member of an infinite family of iteration functions, called the Basic Family in Kalantari et al. (1997). are repulsive. This generalizes a result of Vrscay and Gilbert (1988) who prove the property only for the second member of the family which coincides with the well-known Halley's method. Our result implies that a convergent orbit corresponding to any specific member of the Basic Family will necessarily converge to a zero of f. The Basic Family is a fundamental family with several different representations. It has been rediscovered by several authors using various techniques. The earliest derivation of this family is from an analysis of Schroder (1870). But in fact the Basic Family and its multipoint versions are all derivable from a determinantal generalization of Taylor's theorem (Kalantari (1997)).