New family of eighth‐order methods for nonlinear equation

Abstract

Purpose

The purpose of this paper is to present a new family of iterative methods with eighth‐order convergence for solving the nonlinear equation.

Design/methodology/approach

The paper uses a family of eighth‐order iterative methods for solving nonlinear equation based on Kou's seventh‐order method.

Findings

This family of methods is preferable to Ostrowski's, Grau's and Kou's methods in high‐precision computations.

Research limitations/implications

This paper only deals with the nonlinear equations.

Practical implications

This paper is concerned with the iterative methods for finding a simple root of the nonlinear equation f(x)=0. One of the reasons for discussing the solution of nonlinear equation is that many methods for high‐dimensional optimization problems involve solving a sub‐problem which is a one‐dimensional search problem. And the nonlinear finite element problem, the boundary‐value problems appearing in Kinetic theory of gases, elasticity and other applied areas are also reduced to solving such an equation.

Originality/value

New methods of this family require three evaluations of the function and one evaluation of its first derivative and without using the second derivatives per iteration. This new family of methods as a new example agrees with Kung‐Traub's conjecture for n=4 and achieves its optimal convergence order 2ⁿ⁻¹.

Citation

Zhang, G., Zhang, Y. and Ding, H. (2009), "New family of eighth‐order methods for nonlinear equation", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 28 No. 6, pp. 1418-1427. https://doi.org/10.1108/03321640910991985