Stability of a Family of Iterative Methods of Fourth-Order

• Alicia Cordero
• Lucía Guasp
• Juan R. Torregrosa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)

Abstract

In this paper, we analyze the dynamical anomalies of a family of iterative methods, for solving nonlinear equations, designed by using weight function procedure. All the elements of the family are optimal schemes (in the sense of Kung-Traub conjecture) of fourth-order, but not all have the same stability properties. So, we describe the dynamical behavior of this family on quadratic polynomials. The study of fixed points and their stability, joint with the critical points and their associated parameter planes, show the richness of the presented class and allow us to select the members of the family with good stability properties.

Keywords

Nonlinear equation Iterative method Dynamical behavior Fatou and Julia sets Basin of attraction

Notes

Acknowledgement

This research was partially supported by Ministerio de Economía y Competitividad under grants MTM2014-52016-C2-2-P and Generalitat Valenciana PROMETEO/2016/089.

References

1. 1.
Amat, S., Busquier, S.: Advances in Iterative Methods foir Nonlinear Equations. SIMAI. Springer, Switzerland (2016)
2. 2.
Blanchard, P.: The dynamics of Newton’s method. In: Proceedings of Symposium Applied Mathematics, vol. 49, pp. 139–154 (1994)Google Scholar
3. 3.
Blanchard, P.: Complex analytic dynamics on the Riemann sphere. Bull. AMS 11(1), 85–141 (1984)
4. 4.
Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameter planes of iterative families and methods. Sci. World J. 2013 (2013). Article ID 780153Google Scholar
5. 5.
Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Math. 21, 634–651 (1974)
6. 6.
Petković, M., Neta, B., Petković, L.D., Džunić, J.: Multipoint Methods for Solving Nonlinear Equations. Academic Press, Amsterdam (2013)