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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

McMullen's root-finding algorithm for cubic polynomials


Author: Jane M. Hawkins
Journal: Proc. Amer. Math. Soc. 130 (2002), 2583-2592
MSC (2000): Primary 37F10, 37D20; Secondary 49M99
Published electronically: April 22, 2002
MathSciNet review: 1900865
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. A. Beardon, Iterations of Rational Functions, Springer-Verlag GTM 132, (1991). MR 92j:30026
  • 2. K. Kneisl, Julia sets for the supernewton method, Cauchy's method, and Halley's method, Chaos 11 (2001), 359-370.
  • 3. C. McMullen, Families of rational maps and iterative root-finding algorithms, Ann. of Math. Ser. 2, 125, No.3, (1987), 467-493. MR 88i:58082
  • 4. J. Milnor, Dynamics in One Complex Variable, Vieweg (1999). CMP 2000:03
  • 5. M. Shub and S. Smale, On the existence of generally convergent algorithms, J. of Complexity 2, (1986), 2-11. MR 89d:65054
  • 6. S. Smale, On the efficiency of algorithms of analysis, Bull AMS 13, (1985), 87-121. MR 86m:65061
  • 7. N. Steinmetz, Rational Iteration: Complex Dynamical Systems, de Gruyter Studies in Math 16, (1993). MR 94h:30035
  • 8. E. Vrscay and W. Gilbert, Extraneous fixed points, basin boundaries and chaotic dynamics for Schröder and König rational iteration functions, Numer. Math. 52, 1-16 (1988). MR 89b:30026

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Additional Information

Jane M. Hawkins
Affiliation: Department of Mathematics, CB #3250, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599
Email: jmh@math.unc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06659-5
PII: S 0002-9939(02)06659-5
Keywords: Root-finding algorithms, complex dynamics
Received by editor(s): January 11, 2001
Published electronically: April 22, 2002
Communicated by: Michael Handel
Article copyright: © Copyright 2002 American Mathematical Society