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Attracting cycles for the relaxed Newton’s method. (English) Zbl 1215.65089
Authors’ abstract: We study the relaxed Newton’s method applied to polynomials. In particular, we give a technique such that for any $n\geq 2$, 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 $p(z)=z^m - c$ (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.

65H04Roots of polynomial equations (numerical methods)
65E05Numerical methods in complex analysis
30C15Zeros of polynomials, etc. (one complex variable)
Full Text: DOI
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