Efficient two-step derivative-free iterative methods with memory and their dynamics

Abstract

In this paper, we present a family of three-parameter derivative-free iterative methods with and without memory for solving nonlinear equations. The convergence order of the new method without memory is four requiring three functional evaluations. Based on the new fourth-order method without memory, we present a family of derivative-free methods with memory. Using three self-accelerating parameters, calculated by Newton interpolatory polynomials, the convergence order of the new methods with memory are increased from 4 to 7.0174 and 7.5311 without any additional calculations. Compared with the existing methods with memory, the new method with memory can obtain higher convergence order by using relatively simple self-accelerating parameters. Numerical comparisons are made with some known methods by using the basins of attraction and through numerical computations to demonstrate the efficiency and the performance of the presented methods.

Keywords:

• derivative-free method
• nonlinear equations
• convergence order
• root-finding
• multipoint method with memory

2010 AMS Subject Classifications:

• 65H05
• 65B99

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This study is partly supported by the National Natural Science Foundation of China [No. 11071033], the Ph.D. Start-up Foundation of Liaoning Province of China [Nos. 20141137 and 20141139], the Liaoning BaiQianWan Talents Program [No. 2013921055] and the Educational Commission Foundation of Liaoning Province of China [Nos. L2014444 and L2015012].