A General Way to Construct a New Optimal Scheme with Eighth-Order Convergence for Nonlinear Equations

Abstract

In this paper, we present a new and interesting optimal scheme of order eight in a general way for solving nonlinear equations, numerically. The beauty of our scheme is that it is capable of producing further new and interesting optimal schemes of order eight from every existing optimal fourth-order scheme whose first substep employs Newton’s method. The construction of this scheme is based on rational functional approach. The theoretical and computational properties of the proposed scheme are fully investigated along with a main theorem which establishes the order of convergence and asymptotic error constant. Several numerical examples are given and analyzed in detail to demonstrate faster convergence and higher computational efficiency of our methods.

A General Optimal Iterative Scheme with Arbitrary Order of Convergence

“…The expression of this index, I = p 1/d , involves the order of convergence of the method p and the number of functional evaluations in each iteration, d. The methods with higher convergence order are not necessarily the most efficient, as the complexity of the iterative expression also plays an important role. In [4], Kung and Traub established that the order of convergence of an iterative method without memory, which used n functional evaluations per iteration, is less than or equal to 2 n−1 . When an iterative scheme reaches this upper bound, it is said to be an optimal method.…”

“….. This method is not optimal following the Kung-Traub conjecture [4]. We modify expression (2) to obtain an optimal iterative scheme of order 2 n+1 .…”

Families of Means-Based Modified Newtons Method for Solving Nonlinear Model

“…In order to improve the CO and EI of (1.2), many authors have used different techniques. A good review on some of these techniques such as the geometric, functional, composition, sampling, Adomain decomposition, rational function and weight function techniques can be found in [1,15,16,17] and reference therein. The Weerakoon and Fernando (W-F) method in [25] is one modified form of the NM (1.2), put forward by replacing the function f (x) in (1.2) with the arithmetic mean of the functions f (x) and f (w).…”