Improved Chebyshev–Halley methods with sixth and eighth order convergence

Abstract

We present a second derivative free family of modified Chebyshev–Halley methods with increasing order of convergence for solving nonlinear equations. The idea is based on the recent development by Li et al. (2014). Analysis of convergence shows that the family possesses at least sixth order of convergence. In a particular case even eighth order of convergence is achieved. Per iteration each method of the family requires four evaluations, namely three functions and one first derivative. That means, the eighth order method is optimal in the sense of Kung–Traub hypothesis. Numerical tests are performed, which confirm the theoretical results regarding order of convergence and efficiency of the new methods.

Introduction

The construction of fixed point iterative methods for solving the nonlinear equation $f(x)=0$ is an important and interesting task in numerical analysis and many applied scientific disciplines. A great importance of this subject has led to the development of many iterative methods. Although iterative methods were extensively studied in Traub’s book [1] and some books and papers published in the 1960s, 70s and 80s (see, e.g., [2], [3], [4], [5], [6], [7], [8], [9]). The interest for these methods has renewed in recent years due to the rapid development of digital computers, advanced computer arithmetics and symbolic computation.

Traub [1] has divided iterative methods into two classes, viz. one-point iterative methods and multipoint iterative methods. The important features of these classes of methods are order of convergence and computational efficiency. Order of convergence shows the speed with which a given sequence of iterates converges to the root while the computational efficiency concerns with the economy of the entire process and is defined by $E=p^{1/n}$ (see [2]), where p is the order of convergence and n is the total number of function evaluations required per iteration. Investigation of one-point iterative methods has demonstrated theoretical restrictions on the order and efficiency of these methods (see [1]). However, Kung and Traub [5] have conjectured that multipoint iterative methods without memory based on n evaluations have optimal order $2^{n-1}$. For example, with three function evaluations a two-point method of optimal fourth order convergence can be constructed (see [2], [3], [4], [10], [11]) and with four function evaluations a three-point method of optimal eighth order convergence can be developed [12], [13], [14]. A more extensive list of references as well as a survey on progress made on the class of multipoint methods may be found in the recent book by Petković et al. [15].

The most basic method in the class of one-point methods is the Newton’s method [2] which is given as

$$x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}\text{,}n=0\text{,}1\text{,}2\text{,}\dots \text{,}$$

where $x_0$ is the initial guess to a root. This method has quadratic convergence and requires two evaluations per iteration, namely f and $f^{\prime }$. In order to improve the convergence of Newton’s method, researchers have also proposed cubically convergent one-point methods. One such scheme is the well-known family of Chebyshev–Halley methods [16], which is defined as

$$x_{n+1}=x_n-\left(1+\frac{1}{2}\frac{L_f(x_n)}{1-\alpha L_f(x_n)}\right)\frac{f(x_n)}{f^{\prime }(x_n)}\text{,}$$

where $\alpha \in \mathbb{R}$ and $L_f(x_n)=\frac{f^{″}(x_n)f(x_n)}{f^{\prime }{(x_n)}^2}$. Per iteration, the formula requires three evaluations, namely one each of $f\text{,}{f}^{\prime }$ and $f^{″}$. This family includes the classical Chebyshev’s method for $\alpha =0$, Halley’s method for $\alpha =1/2$ and super-Halley method for $\alpha =1$. For details of these methods, the readers are referred to [17], [18], [19]. It is quite clear that the practical application of Chebyshev–Halley type methods is restricted in the problems where second derivative is difficult to evaluate. This fact has motivated many researchers to explore second derivative free two-point variants of Chebyshev–Halley methods with Newton’s or Newton-like iteration as the first step, see [20], [21], [22], [23] and references therein. Majority of such variants possess the third order convergence. Some modifications, however, achieve optimal fourth order convergence [22], [23].

Based on second derivative free two-point modified Chebyshev–Halley methods, researchers have also proposed three-point methods. For example, Li et al. [24] have recently considered the following scheme of modified Chebyshev–Halley methods

$$\left\{\begin{array}{c}{y}_n=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}\text{,}\hfill \\ z_n=x_n-\left(1+\frac{f(y_n)}{f(x_n)-2\alpha f(y_n)}\right)\frac{f(x_n)}{f^{\prime }(x_n)}\text{,}\hfill \\ x_{n+1}=x_n-\frac{f(z_n)}{f^{\prime }(z_n)}\text{.}\hfill \end{array}\right.$$

As this scheme requires five function evaluations, so to reduce the number of function evaluations Li et al. have used the following approximation of the derivative at the point $z_n$

$$f^{\prime }(z_n)=f^{\prime }(x_n)+{\tilde{f}}^{″}(x_n)(z_n-x_n)\text{,}$$

where ${\tilde{f}}^{″}(x_n)=\frac{2f(y_n)f^{\prime }{(x_n)}^2}{f{(x_n)}^2}$. It has been shown that this scheme has fifth order convergence when $\alpha \ne 1$. For $\alpha =1$, the order is six. It is easy to see that the scheme (3) on using (4) requires four function evaluations, viz. three f and one $f^{\prime }$. However, with the same number of evaluations a more higher order scheme (even a scheme of optimal eighth order) can be achieved. This is the main motivation of present work.

In this paper we consider the modified Chebyshev–Halley scheme (3) and present a one-parameter family of methods with sixth order of convergence. Furthermore, it is shown that a particular value of the used parameter produces eighth order convergence. The family consumes same number of function evaluations as that of the method by Li et al. In this way the eighth order method of the family possesses optimal convergence according to Kung–Traub hypothesis [5]. In Section 2, the method with analysis of convergence is presented. The theoretical results proved in Section 2 are verified in Section 3 by considering various numerical examples. Section 4 includes the concluding remarks.