A new iteration method with cubic convergence to solve nonlinear algebraic equations

doi:10.1016/j.amc.2005.08.020

Applied Mathematics and Computation

Volume 175, Issue 2, 15 April 2006, Pages 1147-1155

https://doi.org/10.1016/j.amc.2005.08.020 Get rights and content

Abstract

In this paper, a new iteration scheme is proposed to solve the roots of an algebraic equation f(x) = 0. Given an initial guess, x₀, the roots of the equation can be obtained using the following iteration scheme: $x_{n + 1} = x_{n} + \frac{- f^{'} (x_{n}) \pm \sqrt{f^{' 2} (x_{n}) - 2 f (x_{n}) f^{″} (x_{n})}}{f^{″} (x_{n})} .$ This iteration scheme has unique convergence characteristics different from the well-known Newton’s method. It is shown that this iteration method has cubic local convergence in the neighborhood of the root. Using this scheme, real or complex roots for specific algebraic equations can be found. Because there are two iteration directions, for a given initial guess, two solutions can be found for certain algebraic equations with multiple roots. Examples are presented and compared with other methods.

Introduction

The techniques to find the roots of algebraic equations have lots of applications in many science or engineering problems. Among the techniques, iteration methods are very popular and are used by many researchers. One of the most well-known iteration methods is called Newton’s method [1], [2], [3]. Other methods such as bisection method, fixed point iteration, secant method are also used in solving algebraic equations [1], [2], [3]. A new iteration method based on Taylor’s theorem was proposed to solve nonlinear algebraic equations by He [4]. It is claimed that this new method has better convergent characteristics than the well-known Newton’s method. However, some problems for the proposed scheme were pointed out in a recent paper [5]. By comparing with Newton’s method, it was shown that the He’s scheme converges slower than Newton’s method and even diverges for some cases. In the current work, a new iteration scheme will be proposed to correct the mentioned problems. The advantages of this new proposed scheme will be discussed and compared with other methods.

Section snippets

Theoretical background

The current approach is based on Taylor’s series expansion. Consider an arbitrary algebraic equation, f(x) = 0. The Taylor’s series expansion around a given initial point x = x₀, assuming x₀ being close enough to the root x = p, is given as follows: $f (x) = f (x_{0}) + f^{'} (x_{0}) (x - x_{0}) + \frac{1}{2} f^{″} (x_{0}) (x - x_{0})^{2} + HOT,$ where HOT denotes the higher order terms. Then the algebraic equation becomes $f (x) = f (x_{0}) + f^{'} (x_{0}) (x - x_{0}) + \frac{1}{2} f^{″} (x_{0}) (x - x_{0})^{2} + HOT = 0 .$ Substituting x = p into Eq. (2) yields $f (p) = f (x_{0}) + f^{'} (x_{0}) (p - x_{0}) + \frac{1}{2} f^{″} (x_{0}) (p - x_{0})^{2} + HOT = 0 .$ When x₀

Convergence analysis

In order to analyze the convergent characteristics of the iteration scheme in Eq. (6), consider the following iteration function: $Ψ (x) = x + \frac{- f^{'} (x) \pm \sqrt{f^{' 2} (x) - 2 f (x) f^{″} (x)}}{f^{″} (x)} .$ In the following derivation, it is assumed that f″(p) ≠ 0. From Eq. (7), it is obtained $Ψ^{'} (x) = \frac{\mp f (x) f^{‴} (x)}{f^{″} (x) \sqrt{f^{' 2} (x) - 2 f (x) f^{″} (x)}} + \frac{f^{‴} (x) [f^{'} (x) \mp \sqrt{f^{' 2} (x) - 2 f (x) f^{″} (x)}]}{f^{″ 2} (x)}$ and $Ψ^{″} (x) = \frac{f^{'} f^{‴} + {ff}^{(4)}}{f^{″}} \frac{\mp 1}{\sqrt{f^{' 2} (x) - 2 f (x) f^{″} (x)}} + \frac{{ff}^{‴ 2}}{f^{' 2}} \frac{\pm 1}{\sqrt{f^{' 2} (x) - 2 f (x) f^{″} (x)}} + \frac{\mp f^{2} f^{‴ 2} (x) [f^{' 2} (x) - 2 f (x) f^{″} (x)]^{- 3 / 2}}{f^{″} (x)} + \frac{f^{‴}}{f^{' 2}} [f^{″} \pm \frac{{ff}^{″}}{\sqrt{f^{' 2} (x) - 2 f (x) f^{″} (x)}}] + [f^{'} \mp \sqrt{f^{' 2} (x) - 2 f (x) f^{″} (x)}] [\frac{f^{(4)}}{f^{' 2}} - \frac{2 f^{″} f^{‴ 2}}{f^{″ 4}}] .$

Some examples

Example 1

As pointed out by Luo [5], He’s scheme fails in an example as f(x) = x³ − e^−x = 0. We will apply the current scheme to this problem. All the examples in the current work are computed using MATLAB with double precision. The first two derivatives can be obtained as $f^{'} (x) = 3 x^{2} + e^{- x}$ and $f^{″} (x) = 6 x - e^{- x}$ We start our iteration from p₀ = 0. The iteration results are shown in Table 1, Table 2 for different signs in the iteration scheme. When using the positive sign in Eq. (6), the result converges to a real root after

Conclusion

In this paper, a new iteration scheme is proposed to solve the roots of algebraic equations. Results show that this method has better performance than the well-known Newton’s method. Unique characteristics of the new method include:

(1)
The new scheme has cubic local convergence in the neighborhood of the root.
(2)
Using one initial guess, two solutions can be found using different iteration directions.
(3)
Given a real initial guess, complex solutions can be obtained.
(4)
Current approach can provide a root

References (6)

J.-H. He
A new iteration method for solving algebraic equations
Appl. Math. Comput.
(2003)
M. Frontini et al.
Some variant of Newton’s method with third-order convergence
Appl. Math. Comput.
(2003)
R.L. Burden et al.
Numerical Analysis
(2001)

There are more references available in the full text version of this article.

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