Estimating convergence regions of Schröder’s iteration formula: how the Julia set shrinks to the Voronoi boundary View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

2018-09-19

AUTHORS

Tomohiro Suzuki, Hiroshi Sugiura, Takemitsu Hasegawa

ABSTRACT

Schröder’s iterative formula of the second kind (S2 formula) for finding zeros of a function f(z) is a generalization of Newton’s formula to an arbitrary order m of convergence. For iterative formulae, convergence regions of initial values to zeros in the complex plane z are essential. From numerical experiments, it is suggested that as order m of the S2 formula grows, the complicated fractal structure of the boundary of convergence regions gradually diminishes. We propose a method of estimating the convergence regions with the circles of Apollonius to verify this result for polynomials f(z) with simple zeros. We indeed show that as m grows, each region surrounded by the circles of Apollonius monotonically enlarges to the Voronoi cell of a zero of f(z). Numerical examples illustrate convergence regions for several values of m and some polynomials. More... »

PAGES

1-17

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s11075-018-0598-8

DOI

http://dx.doi.org/10.1007/s11075-018-0598-8

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1107079974


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