Modified Chebyshev’s method free from second derivative for non-linear equations

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Abstract

In this paper, we present some new modifications of Chebyshev’s method free from second derivative. The order of convergence of the new methods is three. Per iteration the methods require two evaluations of the function and one evaluation of its first derivative. The convergence of the new methods for the case of multiple roots is also discussed. Numerical examples show that the new methods have definite practical utility.

Introduction

Solving non-linear equations is one of the most important problems in numerical analysis. In this paper, we consider iterative methods to find the roots of non-linear equations f(x) = 0, where f:DRR for an open interval D is a scalar function.

Newton’s method (NM) for a single non-linear equation is written asxn+1=xn-f(xn)f(xn).This is an important and basic method [1], which converges quadratically.

Chebyshev’s method [2], [3], known as its order of convergence three, is written asxn+1=xn-1+12f(xn)f(xn)f(xn)2f(xn)f(xn).

However, Chebyshev’s method depends on the second derivatives in computing process and its practical application is restricted rigorously, so that Newton’s method is frequently used to solve non-linear equations because of higher computational efficiency.

For this reason, a family of iterative methods with free second derivative is developedxn+1=xn-1-12θf(yn)-f(xn)f(xn)f(xn)f(xn),by replacing the second derivative of Chebyshev’s method with a finite difference between first derivatives, i.e.f(xn)f(yn)-f(xn)yn-xn,where θR, θ  0 and yn = xn  θ f(xn)/f′(xn). The cases θ = 1/2, θ  (0, 1] and θ < 0 are considered in [4], [5], [6], respectively. These methods are important and interesting because they do not require the second derivative although they can converge cubically.

Recently, another approach is used to remove the second derivative from Halley’s method and some third-order iterative methods free from second derivative are obtained [7].

Here, we will apply the approach in [7] to Chebyshev’s method and obtain a family of modifications of Chebyshev’s method free from second derivative. Analysis of convergence shows that this family of methods is cubically convergent for the case of simple roots. Per iteration these methods require two evaluations of the function and one of its first derivative. The convergence for the the case of multiple roots is also considered. Numerical examples show that the new methods have the definite practical utility.

Section snippets

The methods

Let yn = xn  θf(xn)/f′(xn), where θ is a nonzero real parameter. We consider Taylor expansion of f(yn) about xnf(yn)f(xn)+f(xn)(yn-xn)+12f(xn)(yn-xn)2,which impliesf(yn)(1-θ)f(xn)+12θ2f(xn)f(xn)2f(xn)2.We can now approximate [7]12f(xn)f(xn)f(xn)2f(yn)+(θ-1)f(xn)θ2f(xn).Using (5) in (2), we obtainxn+1=xn-f(yn)+(θ2+θ-1)f(xn)θ2f(xn),where θR, θ  0 and yn = x n  θ f(xn)/f′(xn). In Section 3, we will prove that the methods defined by (6) are cubically convergent for any nonzero real number θ

Analysis of convergence: the case of simple roots

Theorem 1

Assume that the function f:DRR has a simple root x  D, where D is an open interval. If f(x) is sufficiently smooth in the neighborhood of the root x, then the order of convergence of the methods defined by (6) is three.

Proof

Let en = xn  x and dn = yn  x, where yn = xn  θf(x n)/f′(xn).

Using Taylor expansion and taking into account f(x) = 0, we havef(xn)=f(x)[en+c2en2+c3en3+O(en4)],where ck = (1/k!)f(k)(x)/f′(x),k = 2, 3 , …. Furthermore, we havef(xn)=f(x)[1+2c2en+3c3en2+O(en3)].Dividing (11) by (12) gives

Analysis of convergence: the case of multiple roots

Now, we consider the case of multiple roots. Letxn+1=xn-f(yn)+(θ2+θ-1)f(xn)θ2f(xn)=g(xn),where θR, θ  0 and yn = x n  θ f(xn)/f′(xn). The convergence factor of g(xn) is given in the following theorem.

Theorem 2

If x is a root of f(x) with multiplicity p > 1, namely f(x) = 0, f(x) = 0,  , f(p1)(x) = 0, f(p)(x)  0, theng(x)=1-(1-θ/p)p+θ2+θ-1θ2p.

Proof

If x is a root of f(x) with multiplicity p > 1, then we may writef(x)=(x-x)ph(x),with h(x)  0. The first derivative of f(x) is thenf(x)=(x-x)p-1[ph(x)+(x-x)h(x)].

Numerical examples

Now, we employ some new modifications of Chebyshev’s method, Eqs. (8), (10), obtained in this paper to solve some non-linear equations and compare them with Newton’s method (NM), Eq. (1), the method of Potra and Pták [8] (PPM), Eq. (7), and the method obtained in [9], Eq. (9). All computations are carried out with double arithmetic precision. Displayed in Table 2 are the number of iterations (n) and the number of function evaluations (NFE) required such that ∣f(xn)∣ < 1.E  15.

All numerical results

Conclusions

We have shown that it is possible to obtain many modifications of Chebyshev’s method free from second derivative by using an approach to remove the second derivative from Chebyshev’s method. From Theorem 1, we prove that the order of convergence of these methods is three for the case of simple roots. Analysis of efficiency shows that these methods are preferable to the well-known Newton’s method, especially in the case where the computational costs of the first derivative are equal or more than

Acknowledgement

Work supported by National Natural Science Foundation of China (50379038).

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