Attracting cycles for the relaxed Newton's method
- Plaza, S. 1
- Romero, N. 2
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1
Universidad de Santiago de Chile
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2
Universidad de La Rioja
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ISSN: 0377-0427
Year of publication: 2011
Volume: 235
Issue: 10
Pages: 3238-3244
Type: Article
More publications in: Journal of Computational and Applied Mathematics
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Abstract
We study the relaxed Newton's method applied to polynomials. In particular, we give a technique such that for any n<2, we may construct a polynomial so that when the method is applied to a polynomial, the resulting rational function has an attracting cycle of period n. We show that when we use the method to extract radicals, the set consisting of the points at which the method fails to converge to the roots of the polynomial p(z)=zm-c (this set includes the Julia set) has zero Lebesgue measure. Consequently, iterate sequences under the relaxed Newton's method converge to the roots of the preceding polynomial with probability one. © 2011 Elsevier B.V. All rights reserved.