Fourth-order convergent iterative method for nonlinear equation

doi:10.1016/j.amc.2006.04.068

Applied Mathematics and Computation

Volume 182, Issue 2, 15 November 2006, Pages 1149-1153

https://doi.org/10.1016/j.amc.2006.04.068 Get rights and content

Abstract

In this paper, we suggest and analyze a new iterative method for finding approximate solution of the nonlinear equation f(x) = 0. It is shown that proposed method has fourth-order convergence. Several numerical examples are given to illustrate that the method developed in this paper give better results than the other methods including Newton method.

Introduction

In recent years, several iterative methods have been developed for finding the numerical solutions of nonlinear equation f(x) = 0, see [1], [2], [3], [4], [5], [6], [7], [8] and the reference therein. Inspired and motivated by the research going on in this area, we suggest a new iterative method which has quadratic convergence. Combining this new method and Newton’s method, we consider a predictor–corrector method for solving nonlinear equation f(x) = 0. Several numerical examples are given to illustrate the efficiency of these new methods.

We have shown that new method can be applied in some cases when the Newton method fails to give desired result.

Section snippets

Iterative methods

Now we will consider the nonlinear equation $f (x) = 0 .$

Let r be the exact root and x₀ be the initial guess known for the required root. Assume $x_{1} = x_{0} + h, h ≪ 1,$ be the first approximation to the root.

Consider the following auxiliary equation with a parameter p: $g (x) = p^{3} (x - x_{0})^{2} f (x)^{2} - f (x) = 0,$ where p ∈ R and ∣p∣ < ∞.

It is clear that the root of (2.1) is also the root of (2.3) and vice versa. If x₁ = x₀ + h is the better approximation for the required root, then (2.3) gives $p^{3} h^{2} f^{2} (x_{0} + h) - f (x_{0} + h) = 0 .$

Expending by the Taylor’s

Convergence analysis

In this section, we consider the convergence analysis of iterative technique given in Algorithm 2.2.

Theorem 3.1

Let r ∈ I be a simple zero of sufficiently differentiable function f : I ⊆ R → R for an open interval I. If x₀ is sufficiently close to r, then the iterative method defined by Algorithm 2.2 has fourth-order convergence.

Proof

The technique is given by $y_{n} = x_{n} - \frac{2 f (x_{n})}{f^{'} (x_{n}) \pm \sqrt{f^{' 2} (x_{n}) + 4 p^{3} f^{3} (x_{n})}},$ $x_{n + 1} = y_{n} - \frac{f (y_{n})}{f^{'} (y_{n})} .$ From (3.1), we get $x_{n + 1} = x_{n} - \frac{f (x_{n})}{f^{'} (x_{n}) [1 + p^{3} \frac{f^{3} (x_{n})}{f^{' 2} (x_{n})}]} .$ Let r be a simple zero of f. Since f is

Numerical examples

We present some examples to illustrate the efficiency of the new developed predictor–corrector iterative methods in this paper. We compare the Newton method (NM), the method of Abbasbandy [1] (AM), the method of Chun [3] (CM) and NA (Algorithm 2.2), the method introduced in this present paper. We used ε = 10⁻¹⁵. The following stopping criteria are used for computer programs: $\begin{matrix} (i) | x_{n + 1} - x_{n} | < ε, \\ (ii) | f (x_{n + 1}) | < ε . \end{matrix}$ The examples are the same as in Chun [3]. $\begin{matrix} f_{1} (x) = \sin^{2} x - x^{2} + 1, \\ f_{2} (x) = x^{2} - e^{x} - 3 x + 2, \\ f_{3} (x) = \cos x - x, \\ f_{4} (x) = \end{matrix}$

Acknowledgement

This research is supported by the Higher Education Commission, Pakistan, through Research Grant No. i-28/HEC/HRD/2005/90.

References (8)

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C. Chun
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X. Wu et al.
On a lass of quadratic convergence iteration formulae without derivatives
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Richard L. Burden et al.
Numerical Analysis
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There are more references available in the full text version of this article.

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