McMullen’s root-finding algorithm for cubic polynomials
HTML articles powered by AMS MathViewer
- by Jane M. Hawkins PDF
- Proc. Amer. Math. Soc. 130 (2002), 2583-2592 Request permission
Abstract:
We show that a generally convergent root-finding algorithm for cubic polynomials defined by C. McMullen is of order 3, and we give generally convergent algorithms of order 5 and higher for cubic polynomials. We study the Julia sets for these algorithms and give a universal rational map and Julia set to explain the dynamics.References
- Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR 1128089, DOI 10.1007/978-1-4612-4422-6
- K. Kneisl, Julia sets for the supernewton method, Cauchy’s method, and Halley’s method, Chaos 11 (2001), 359-370.
- Curt McMullen, Families of rational maps and iterative root-finding algorithms, Ann. of Math. (2) 125 (1987), no. 3, 467–493. MR 890160, DOI 10.2307/1971408
- J. Milnor, Dynamics in One Complex Variable, Vieweg (1999).
- Michael Shub and Steve Smale, On the existence of generally convergent algorithms, J. Complexity 2 (1986), no. 1, 2–11. MR 925341, DOI 10.1016/0885-064X(86)90020-8
- Steve Smale, On the efficiency of algorithms of analysis, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 2, 87–121. MR 799791, DOI 10.1090/S0273-0979-1985-15391-1
- Norbert Steinmetz, Rational iteration, De Gruyter Studies in Mathematics, vol. 16, Walter de Gruyter & Co., Berlin, 1993. Complex analytic dynamical systems. MR 1224235, DOI 10.1515/9783110889314
- Edward R. Vrscay and William J. Gilbert, Extraneous fixed points, basin boundaries and chaotic dynamics for Schröder and König rational iteration functions, Numer. Math. 52 (1988), no. 1, 1–16. MR 918313, DOI 10.1007/BF01401018
Additional Information
- Jane M. Hawkins
- Affiliation: Department of Mathematics, CB #3250, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599
- MR Author ID: 82840
- Email: jmh@math.unc.edu
- Received by editor(s): January 11, 2001
- Published electronically: April 22, 2002
- Communicated by: Michael Handel
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 2583-2592
- MSC (2000): Primary 37F10, 37D20; Secondary 49M99
- DOI: https://doi.org/10.1090/S0002-9939-02-06659-5
- MathSciNet review: 1900865