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Julia sets for the super-Newton method, Cauchy’s method, and Halley’s method. (English) Zbl 1080.65532
Summary: We study numerically and dynamically three cubically convergent iterative root-finding algorithms, namely Cauchy’s method, the super-Newton method, and Halley’s method. Using the concept of a universal Julia set (motivated by the results of McMullen), we establish that these algorithms converge when applied to any quadratic with distinct roots. We give examples showing the existence of attracting periodic orbits not associated to a root for the super-Newton method and Halley’s method applied to cubic polynomials. We include computer plots showing the dynamic structure for each algorithm applied to a variety of polynomials.
MSC:
65H05Single nonlinear equations (numerical methods)
30C15Zeros of polynomials, etc. (one complex variable)
37F50Small divisors, rotation domains and linearization; Fatou and Julia sets