Attracting cycles for the relaxed Newton’s method

✍ Scribed by Sergio Plaza; Natalia Romero

• Book ID: 104007311
• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 491 KB
• Volume: 235
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2011.01.010

✦ Synopsis

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) = 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.