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McMullen's root-finding algorithm for cubic polynomials
Author(s):
Jane
M.
Hawkins
Journal:
Proc. Amer. Math. Soc.
130
(2002),
2583-2592.
MSC (2000):
Primary 37F10, 37D20;
Secondary 49M99
Posted:
April 22, 2002
MathSciNet review:
1900865
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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:
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- 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:
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
Posted:
April 22, 2002
Communicated by:
Michael Handel
Copyright of article:
Copyright
2002,
American Mathematical Society
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