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

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 Th...

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Bibliographic Details
Published in:Applied mathematics and computation 2015-01, Vol.251, p.507-520
Main Authors: Honorato, Gerardo, Plaza, Sergio
Format: Article
Language:English
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
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Summary: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.
ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2014.11.083