Study of the dynamics of third-order iterative methods on quadratic polynomials

Abstract

In this paper, we analyse the dynamical behaviour of the operators associated with multi-point interpolation iterative methods and frozen derivative methods, for solving nonlinear equations, applied on second-degree complex polynomials. We obtain that, in both cases, the Julia set is connected and separates the basins of attraction of the roots of the polynomial. Moreover, the Julia set of the operator associated with multi-point interpolation methods is the same as the Newton operator, although it is more complicated for the frozen derivative operator. We explain these differences by obtaining the conjugacy function of each method and by showing that the operators associated with Newton's method and multi-point interpolation methods are both conjugate to powers of z.

Keywords:

• nonlinear equations
• iterative methods
• complex dynamics
• conjugacy map
• basin of attraction

2010 AMS Subject Classifications:

• 37F10
• 65H05

Acknowledgements

The authors thank Professors X. Jarque and A. Garijo for their help. The authors also thank the referees for their valuable comments and suggestions that have improved the content of this paper. This research was supported by Ministerio de Ciencia y Tecnología MTM2011-28636-C02-02 and by Vicerrectorado de Invetigación, Universitat Politècnica de València, PAID-06-2010-2285