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

Abstract

In literature, the arithmetic mean of the two functions in the denominator of the second step of order three Weerakoon and Fernando (2000) iterative method have been replaced with other different means. However, these actions have not improve its order of convergence. To improve the order of convergence of these modified methods, a generic family of iterative methods that involve two weight functions and a generic consequential function for replacement of means is proposed. The analysis of convergence carried out on the families of methods, shows that they are of fourth order convergence and requires evaluation of three functions per iteration cycle. Further, the flexibility of the weight functions enables the re-discovery of some existing and construction of new families of iterative methods. Some concrete members of the family of methods are applied to solve some nonlinear equations and real life problems that are modeled into nonlinear equations

Means-based class of modified Weerakoon-Fernando method for determining the zero of nonlinear equation

“…Definition 1: An iterative method efficiency is measured using the numerical value obtained by 𝜌 1 𝑇 , where 𝑇 is the sum of all different functions computation in a complete iteration cycle, see Traub, 1964;Ogbereyivwe and Ojo-Orobosa, 2021). ) can be obtained from the iteration function of the method by the use of Taylor's expansions of 𝜇(•) and 𝜇 ′ (•) as contained in the method, then 𝜂 𝑖+1 is referred to as the iterative method error equation, 𝛽 is error equation constant and 𝜌 represents its theoretical CO, see (Traub, 1964;Ogbereyivwe and Ojo-Orobosa, 2021).…”

“…In Ogbereyivwe and Ojo-Orobosa (2021), the authors modified the work of Soleymani et al (2005), by replacing the arithmetic mean 𝑀 𝐴 [•,•] 𝜇 ′ (𝑠 𝑖 ) ). They put forward their modification as:…”

“…Deriving motivation from the use of the MGF utilized in Ogbereyivwe and Ojo-Orobosa (2021) in constructing new iterative method, this article offers a wide class of enhanced WFM in Weerakoon and Fernando (2000), for deciding the zero of nonlinear equations. The new class of methods were developed based on the use of the rational weight function and powermeans generating functions.…”