Dynamics of a fifth-order iterative method

Abstract

In this paper, we study the dynamical behaviour of a two-point iterative method with order of convergence five to solve nonlinear equations in the complex plane. In fact, we complement the dynamical study started in previous works with a more systematic analysis for polynomials with at most two different roots and different multiplicities. In addition, we characterize some polynomials of degree greater or equal to 4, such that the related methods are not generally convergent. We also analyse the degrees of the rational functions associated with two-point methods when they are applied to polynomials of degree n, showing their dependence on n ² and how this fact considerably complicates the dynamical study.

Keywords:

• general convergence
• nonlinear equations
• iterative processes
• Julia sets
• order of convergence

2010 AMS Subject Classifications :

• 30C15
• 37F99
• 37N30

Acknowledgements

This study has been supported by the grant 2011-28636-C02-01, Spanish Ministry of Science and Innovation and by the Fondecyt grant no. 1095025.

Notes

^†Dedicated to the lovely memory of Prof. Sergio Plaza, who unexpectedly died after the submission of this paper.