Novel computational derivative-free methods for simple roots. (English) Zbl 1272.65043

Soleymani, Fazlollah; Soleimani, Farahnaz

Fixed Point Theory 13, No. 1, 247-258 (2012).

Summary: Some novel comptutational techniques for solving single variable nonlinear equations are given. The schemes are without memory and free from derivative evaluations per full iteration. They are built by applying the weight function approach alongside an approximation for the first derivative of the function in the second step of a two-step cycle for obtaining optimal fourth-order schemes; and also by adapting a nonlinear fraction in the stird step of a three-step cycle to attain seventh-order techniques. The classical efficiency indices of the proposed two- and three-step derivative-free methods are 1.587 and 1.626, respectively, up to now. Further research has also been done via the concept of weight functions to provide optimal eighth-order derivative-free techniques which possess 1.682 as their efficiency index. The superiority of the techniques is illustrated by solving numerical examples.

MSC:

65H05	Numerical computation of solutions to single equations
65Y20	Complexity and performance of numerical algorithms

Keywords:

nonlinear equations; efficiency index; optimality; simple root; two-step iterative methods; three-step iterative methods; derivative-free; convergence rate; numerical examples