On improved Steffensen type methods with optimal eighth-order of convergence

Munish Kansal*, V. Kanwar and Saurabh Bhatia

University Institute of Engineering and Technology, Panjab University, Chandigarh 160014, India

E-mail : mkmaths23@pu.ac.in, vmithil@yahoo.co.in, s_bhatia@pu.ac.in

ABSTRACT:

This paper presents an improvement of the existing eighth-order derivative involved method [14] into derivative-free scheme, holding the order of convergence of the original method. Each member of the family requires only four function evaluations per iteration to achieve the eighth-order of convergence, while they are totally free from derivative evaluation. Hence, they agree with the optimality conjecture of Kung-Traub for providing multipoint iterations without memory. The proposed methods are compared with their closest competitors in a series of numerical experiments. Numerical experiments show that such derivative-free, high order schemes offer significant advantages over the derivative involved methods.

KEYWORDS: Nonlinear equations, Steffensen’s method, King’s method, Ostrowski’s method, Efiiciency index, Optimal order of convergence.

Mathematics Subject Classification (2000): 65H05, 65B99

1. INTRODUCTION:

We consider an equation of the form f (x) = 0, where f : D ⊆ R → R be a nonlinear continuous function. Analytical methods for solving such equations are almost non-existent and therefore, it is only possible to obtain approximate solutions by relying on numerical methods based on iterative procedure (see e.g. [1-18]). The classical quadratically convergent Newton’s method is the most celebrated of all one dimensional root-finding methods for solving nonlinear equations, which is given by

Multi-point iterative methods for solving nonlinear equation have drawn a considerable attention in the first decade of the 21st century, which led to the construction of many methods of this type. These methods are primarily introduced with the aim to achieve as high as possible order of convergence using a fixed number of function evaluations. However, multi-point methods do not use higher order derivatives and has great practical importance since they overcome the theoretical limitations of one-point methods regarding their convergence order and computational efficiency.

As the order of an iterative method increases, so does the number of functional evaluations per step. The efficiency index [2] gives a measure of the balance between those quantities, according to the formula p^1/d , where p is the order of convergence of the method and d the number of functional evaluations per step. According to the Kung-Traub conjecture [3], the order of convergence of any multi-point method without memory consuming n functional evaluations cannot exceed the bound 2ⁿ⁻¹, called the optimal order. Thus, the optimal order for a method with three functional evaluations per step would be four. Recently, many researchers have developed the idea of removing derivatives from the iteration function in order to avoid defining new functions such as the first or second derivative, and calculate iterates only by using the function that describes the problem, and obviously trying to preserve the convergence order. In particular, when the first-order derivative , we get a well-known derivative-free method, known as Steffensen method [7] as follows:

where w_n= x_n + f (x_n) and f [·, ·] denotes the first order divided difference. As a matter of fact, both methods maintain quadratic convergence using only two functional evaluations per full step, but Steffensen method is derivative free, which is so useful in optimization problems. Therefore, many higher-order derivative-free methods are built according to the Stefffensen’s method, see [9–12,15–18] and references therein.

Therefore, construction of optimal higher-order multipoint methods which are totally free from derivatives is an open and challenging problem in computational mathematics. With this aim, we intend to propose an optimal eighth-order derivative-free family requiring only four function evaluations, viz., f(x_n), f (w_n), f (y_n), and, f (z_n). As a result, the efficiency index of the proposed scheme is E = 8^1/4≈ 1.681. All the proposed methods considered here are found to be more effective and comparable to the existing robust methods available in literature.

2 The improved derivative-free family with optimal eighthorder of convergence

In this section, we intend to develop optimal eighth-order derivative free scheme. To build a new eighth-order derivative-free optimal family, we consider the following three-step method without memory proposed in [14] with optimal eighth-order convergence:

4. CONCLUSIONS;

We establish a three-step eighth-order Steffensen type methods based on existing derivative involved eighth-order method for finding simple root of a nonlinear equation.The main advantage of these methods is that they do not use the evaluation of any derivatives but a optimal order of convergence is nonetheless maintained. In terms of computational cost, each member of the

family requires only four function evaluations, viz., f (x_n), f (w_n), f (y_n), f (z_n) per iteration to acheive optimal index of efficiency E = 8^1/4≈ 1.682. We have also given a detailed proof to prove the theoretical eighth-order of convergence of the presented families. The numerical experiments suggests that the new class would be valuable alternative for solving nonlinear equations. The numerical and theoretical results show that the new methods have much practical utility.

5. REFERENCES:

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Received on 28.01.2015 Accepted on 12.02.2015

Research J. Engineering and Tech. 6(1): Jan.-Mar. 2015 page 223-228

DOI: 10.5958/2321-581X.2015.00033.1