Numerische Mathematik

, Volume 52, Issue 1, pp 1–16

Extraneous fixed points, basin boundaries and chaotic dynamics for Schröder and König rational iteration functions

  • Edward R. Vrscay
  • William J. Gilbert

DOI: 10.1007/BF01401018

Cite this article as:
Vrscay, E.R. & Gilbert, W.J. Numer. Math. (1987) 52: 1. doi:10.1007/BF01401018


The Schröder and König iteration schemes to find the zeros of a (polynomial) functiong(z) represent generalizations of Newton's method. In both schemes, iteration functionsfm(z) are constructed so that sequenceszn+1=fm(zn) converge locally to a rootz* ofg(z) asO(|znz*|m). It is well known that attractive cycles, other than the zerosz*, may exist for Newton's method (m=2). Asm increases, the iteration functions add extraneous fixed points and cycles. Whether attractive or repulsive, they affect the Julia set basin boundaries. The König functionsKm(z) appear to minimize such perturbations. In the case of two roots, e.g.g(z)=z2−1, Cayley's classical result for the basins of attraction of Newton's method is extended for allKm(z). The existence of chaotic {zn} sequences is also demonstrated for these iteration methods.

Subject classifications

AMS(MOS): 30D05, 30-04, 65E05, 65H05CR: G.1.5

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Edward R. Vrscay
    • 1
  • William J. Gilbert
    • 2
  1. 1.Department of Applied MathematicsUniversity of WaterlooWaterlooCanada
  2. 2.Department of Pure Mathematics, Faculty of MathematicsUniversity of WaterlooWaterlooCanada