Dynamical aspects of some convex acceleration methods as purely iterative algorithm for Newton’s maps

• Published in: Applied mathematics and computation Vol. 251; pp. 507 - 520
• Main Authors: Honorato, Gerardo, Plaza, Sergio
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.01.2015
• Subjects: Computation; Conjugacy classes; Dynamics; Iterative algorithms; Iterative methods; Mathematical models; Newton methods; Polynomials; Rational maps; Roots; The super-Halley iterative method; Theorems; Whittaker’s iterative method; The super-Halley iterative method; Conjugacy classes; Rational maps; Dynamics; Whittaker’s iterative method
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2014.11.083

In this paper we define purely iterative algorithm for Newton’s maps which is a slight modification of the concept of purely iterative algorithm due to Smale. For this, we use a characterization of rational maps which arise from Newton’s method applied to complex polynomials. We prove the Scaling Theorem for purely iterative algorithm for Newton’s map. Then we focus our study in dynamical aspects of three root-finding iterative methods viewed as a purely iterative algorithm for Newton’s map: Whittaker’s iterative method, the super-Halley iterative method and a modification of the latter. We give a characterization of the attracting fixed points which correspond to the roots of a polynomial. Also, numerical examples are included in order to show how to use the characterization of fixed points. Finally, we give a description of the parameter spaces of the methods under study applied to a one-parameter family of generic cubic polynomials.