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

 

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
Published electronically: 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 $ 2\times 2$ matrices. Here we develop an analogous theory based on $ 3\times 3$ matrices which yields a two-variable generally convergent algorithm for cubics.


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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
Published electronically: 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.