Kneisl, Kyle Julia sets for the super-Newton method, Cauchy’s method, and Halley’s method. (English) Zbl 1080.65532 Chaos 11, No. 2, 359-370 (2001). 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. Cited in 41 Documents MSC: 65H05 Numerical computation of solutions to single equations 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) 37F50 Small divisors, rotation domains and linearization in holomorphic dynamics PDFBibTeX XMLCite \textit{K. Kneisl}, Chaos 11, No. 2, 359--370 (2001; Zbl 1080.65532) Full Text: DOI References: [1] Smale S., Bull. Am. Math. Soc. 13 pp 87– (1985) · Zbl 0592.65032 · doi:10.1090/S0273-0979-1985-15391-1 [2] McMullen C., Ann. Math. 125 pp 467– (1987) · Zbl 0634.30028 · doi:10.2307/1971408 [3] Haeseler F. V., Acta Appl. Math. 13 pp 3– (1988) · Zbl 0671.30023 · doi:10.1007/BF00047501 [4] Shub M., J. Complexity 2 pp 2– (1986) · Zbl 0595.65048 · doi:10.1016/0885-064X(86)90020-8 [5] Vrscay E., Numer. Math. 52 pp 1– (1988) · Zbl 0612.30025 · doi:10.1007/BF01401018 [6] Vrscay E., Math. Comput. 46 pp 151– (1986) [7] Halley E., Philos. Trans. R. Soc. London 18 pp 136– (1694) · doi:10.1098/rstl.1694.0029 [8] Hansen E., Numer. Math. 27 pp 257– (1977) · Zbl 0361.65041 · doi:10.1007/BF01396176 [9] Schröder E., Math. Ann. 2 pp 317– (1870) · JFM 02.0042.02 · doi:10.1007/BF01444024 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.