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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 $n\ge 2$, 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 $p\left(z\right)={z}^{m}-c$ (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.
##### MSC:
 65H04 Roots of polynomial equations (numerical methods) 65E05 Numerical methods in complex analysis 30C15 Zeros of polynomials, etc. (one complex variable)