Family Of Eighth-order Methods Research Articles

Eighth order family of iterative methods for nonlinear equations and their basins of attraction

We present a new family of eighth order methods for solving nonlinear equations. The order of convergence of the considered methods is proved and corresponding asymptotic error constants are expressed in terms of four parameters. Numerical examples, obtained using Mathematica with high precision arithmetic, demonstrate convergence and efficacy of our family of methods. For some combinations of parameter values, the new eighth order methods produce very good results on tested examples compared to the results produced by some of the eighth order methods existing in the related literature. An exploration of the relevant dynamics of the proposed methods is presented along with illustrative basins of attraction for various polynomials.

A novel family of weighted-Newton optimal eighth order methods with dynamics

In this paper, we present a family of eighth order methods for solving nonlinear equations. Formula is composed of three steps, namely; Newton iteration in the first step and weighted-Newton iterations in second and third steps. Hence the name weighted-Newton methods. In terms of computational cost, the family requires three evaluations of function and one of first derivative. Therefore, it is optimal in the sense of Kung–Traub conjecture and has efficiency index 1.682 which is better than that of Newton method of efficiency index 1.414 and many other higher order methods. Numerical examples are considered to support that the method thus obtained is competitive with other similar robust methods. Moreover, basins of attraction are presented to demonstrate the performance in complex plane.

Optimal Eighth Order Iterative Methods

We develop an eighth order family of methods, consisting of three steps and three parameters, for solving nonlinear equations. Per iteration the methods require four evaluations (three function evaluations and one evaluation of the first derivative). Convergence analysis shows that the family is eighth-order convergent which is also substantiated through the numerical work. Computational results ascertain that family of methods are efficient and demonstrate equal or better performance as compared with other well known methods.

A new family of modified Ostrowski’s methods with accelerated eighth order convergence

Based on Ostrowski’s fourth order method, we derive a family of eighth order methods for the solution of nonlinear equations. In terms of computational cost the family requires three evaluations of the function and one evaluation of first derivative. Therefore, the efficiency index of the present methods is 1.682 which is better than the efficiency index 1.587 of Ostrowski’s method. Kung and Traub conjectured that multipoint iteration methods without memory based on n evaluations have optimal order 2n − 1. Thus, the family agrees with Kung–Traub conjecture for the case n = 4. The efficacy of the present methods is tested on a number of numerical examples. It is observed that our methods are competitive with other similar robust methods and very effective in high precision computations.

Three-step iterative methods with eighth-order convergence for solving nonlinear equations

A family of eighth-order iterative methods for the solution of nonlinear equations is presented. The new family of eighth-order methods is based on King’s fourth-order methods and the family of sixth-order iteration methods developed by Chun et al. Per iteration the new methods require three evaluations of the function and one evaluation of its first derivative. Therefore this family of methods has the efficiency index which equals 1.682. Kung and Traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2 n − 1 . Thus we provide a new example which agrees with the conjecture of Kung–Traub for n = 4 . Numerical comparisons are made to show the performance of the presented methods.