Summary
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 functionsf m (z) are constructed so that sequencesz n+1 =f m (z n ) converge locally to a rootz * ofg(z) asO(|z n −z *|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 functionsK m (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 allK m (z). The existence of chaotic {z n } sequences is also demonstrated for these iteration methods.
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Vrscay, E.R., Gilbert, W.J. Extraneous fixed points, basin boundaries and chaotic dynamics for Schröder and König rational iteration functions. Numer. Math. 52, 1–16 (1987). https://doi.org/10.1007/BF01401018
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DOI: https://doi.org/10.1007/BF01401018