International Journal of Engineering Mathematics

Volume 2016, Article ID 8565821, 12 pages

http://dx.doi.org/10.1155/2016/8565821

## An Efficient and Straightforward Numerical Technique Coupled to Classical Newton’s Method for Enhancing the Accuracy of Approximate Solutions Associated with Scalar Nonlinear Equations

Aix-Marseille Université, IFSTTAR, LBA UMR_T24, 13016 Marseille, France

Received 3 June 2016; Accepted 14 August 2016

Academic Editor: Yurong Liu

Copyright © 2016 Grégory Antoni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This study concerns the development of a straightforward numerical technique associated with Classical Newton’s Method for providing a more accurate approximate solution of scalar nonlinear equations. The proposed procedure is based on some practical geometric rules and requires the knowledge of the local slope of the curve representing the considered nonlinear function. Therefore, this new technique uses, only as input data, the first-order derivative of the nonlinear equation in question. The relevance of this numerical procedure is tested, evaluated, and discussed through some examples.

#### 1. Introduction

The resolution of nonlinear problems is an issue frequently encountered in several scientific fields such as mathematics, physics, or many engineering branches, for example, mechanics of solids [1–8]. In most cases, these problems are governed by nonlinear equations not having any analytical solution. In this regard, the introduction of iterative methods is therefore needed in order to provide a numerical approximate solution associated with any type of nonlinear equation [9–23]. Among these iterative algorithms, Classical Newton’s Method (CNM) [24, 25] is one of the most used mainly for the following reasons: (i) the simplicity for numerical implementation in any scientific computation software; (ii) the only knowledge of the first-order derivative of the considered function; (iii) the quadratic rate of convergence. In this paper, we propose a New Numerical Technique (NNT) based on geometric considerations which enable providing a more accurate approximate solution than that obtained by CNM. The present study is organized as follows: (i) in the first part, Section 2.1, we outline the scientific framework of this study, then, in second part, Section 2.2, we recall CNM including some convergence results, and, finally, in the third part, Section 2.3, we present NNT which uses only the first-order derivative of the considered nonlinear equation in order to enhance the predictive abilities of CNM; (ii) in Section 3, the numerical relevance of the proposed procedure is addressed, assessed, and discussed on some specific examples.

#### 2. A New Numerical Technique (NNT) Combined with Classical Newton’s Method (CNM)

##### 2.1. Problem Statement

We consider scalar-valued nonlinear equation (with ) in the following form (see Figure 1):where denotes the class of infinitely differentiable functions in domain , is the simple solution (so-called “simple true zero” or “simple root”) on interval , that is, with .