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A root-finding algorithm for cubics
Author:
Sam Northshield
Journal:
Proc. Amer. Math. Soc. 141 (2013), 645-649
MSC (2010):
Primary 65H04; Secondary 26C10, 30D05, 37F10
Posted:
May 24, 2012
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Abstract: Newton's method applied to a quadratic polynomial converges rapidly to a root for almost all starting points and almost all coefficients. This can be understood in terms of an associative binary operation arising from  matrices. Here we develop an analogous theory based on  matrices which yields a two-variable generally convergent algorithm for cubics.
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Jane
M. Hawkins, McMullen’s root-finding
algorithm for cubic polynomials, Proc. Amer.
Math. Soc. 130 (2002), no. 9, 2583–2592 (electronic). MR 1900865
(2003k:37062), http://dx.doi.org/10.1090/S0002-9939-02-06659-5
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Curt
McMullen, Families of rational maps and iterative root-finding
algorithms, Ann. of Math. (2) 125 (1987), no. 3,
467–493. MR
890160 (88i:58082), http://dx.doi.org/10.2307/1971408
- 3.
Sam
Northshield, On two types of exotic addition, Aequationes
Math. 77 (2009), no. 1-2, 1–23. MR 2495716
(2010b:39030), http://dx.doi.org/10.1007/s00010-008-2952-8
- 1.
- J. Hawkins, McMullen's root-finding algorithm for cubic polynomials, Proc. Amer. Math. Soc., 130 (2002), no. 9, 2583-2592. MR 1900865 (2003k:37062)
- 2.
- C. McMullen, Families of rational maps and iterative root-finding algorithms, Ann. of Math. (2) 125 (1987), no. 3, 467-493. MR 890160 (88i:58082)
- 3.
- S. Northshield, On two types of exotic addition, Aequationes Math. 77 (2009), no. 1-2, 1-23. MR 2495716 (2010b:39030)
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Additional Information
Sam Northshield
Affiliation:
Department of Mathematics, State University of New York, Plattsburgh, New York 12901
Email:
northssw@plattsburgh.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-2012-11324-3
PII:
S 0002-9939(2012)11324-3
Keywords:
Newton’s method,
iterative algorithm,
generally convergent.
Received by editor(s):
June 1, 2010
Received by editor(s) in revised form:
June 22, 2011
Posted:
May 24, 2012
Communicated by:
Sergei K. Suslov
Article copyright:
© Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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