Some New Higher Order Weighted Newton Methods for Solving Nonlinear Equation with Applications

Jayakumar Jayaraman,Parimala Sivakumar

doi:10.3390/mca24020059

Jayakumar Jayaraman, Parimala Sivakumar

Open Access

https://doi.org/10.3390/mca24020059

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Abstract

Three new iterative methods for solving scalar nonlinear equations using weight function technique are presented. The first one is a two-step fifth order method with four function evaluations which is improved from a two-step Newton’s method having same number of function evaluations. By this, the efficiency index of the new method is improved from 1.414 to 1.495. The second one is a three step method with one additional function evaluation producing eighth order accuracy with efficiency index 1.516. The last one is a new fourth order optimal two-step method with efficiency index 1.587. All these three methods are better than Newton’s method and many other equivalent higher order methods. Convergence analyses are established so that these methods have fifth, eighth and fourth order respectively. Numerical examples ascertain that the proposed methods are efficient and demonstrate better performance when compared to some equivalent and optimal methods. Seven application problems are solved to illustrate the efficiency and performance of the proposed methods.

Highlights

This paper concerns the numerical solution of nonlinear equations of the general form f ( x ) = 0.Such equations appear in real world problems frequently while there is no closed form solution for them
This method is an optimal method with efficiency index (EI) 1.414
We give some applications and compare the proposed methods to other well known methods: Application 1: We consider the classical projectile problem [23] in which a projectile is launched from a tower of height h > 0, with initial speed v and at an angle φ with respect to the horizontal distance onto a hill, which is defined by the function ω, called the impact function which is dependent on the horizontal distance, x

Summary

Introduction

One of the common problems encountered in science and engineering is: given a single variable function f ( x ), find the values of x for which f ( x ) = 0 The root of such nonlinear equations may be real or complex. The most familiar iterative without memory method is the Newton–Raphson method which is given by ψ2nd N M ( x ) = xn − This method is an optimal method with efficiency index (EI) 1.414. We have presented three new Newton-type iterative methods having fifth, eighth and fourth order convergence whose efficiency indices are 1.495, 1.516 and 1.587 respectively. Among these three methods, fourth order method is a class of optimal method.

Preliminaries

Construction of New Methods

Further Development

Convergence Analysis

Numerical Examples

Some Real Life Applications

Findings

Conclusions