Some variants of the Chebyshev–Halley family of methods with fifth order of convergence

Abstract

In this paper we present some techniques for constructing high-order iterative methods in order to approximate the zeros of a non-linear equation f(x)=0, starting from a well-known family of cubic iterative processes. The first technique is based on an additional functional evaluation that allows us to increase the order of convergence from three to five. With the second technique, we make some changes aimed at minimizing the calculus of inverses. Finally, looking for a better efficiency, we eliminate terms that contribute to the error equation from sixth order onwards.

The paper contains a comparative study of the asymptotic error constants of the methods and some theoretical and numerical examples that illustrate the given results. We also analyse the efficiency of the aforementioned methods, by showing some numerical examples with a set of test functions and by using adaptive multi-precision arithmetic in the computation.

Keywords:

• non-linear equations
• iterative methods
• Chebyshev-Halley method
• order of convergence
• computational efficiency

2000 AMS Subject Classification :

• 65H05

Acknowledgements

This work was supported by the grant MTM2005-03091 and Spanish Ministry of Education and Science.