Numer Algor (2007) 44:381–389 DOI 10.1007/s11075-007-9120-4 O R I G I N A L PA P E R
Modified families of multi-point iterative methods for solving nonlinear equations V. Kanwar & S. K. Tomar
Received: 5 April 2007 / Accepted: 22 June 2007 / Published online: 1 August 2007 # Springer Science + Business Media B.V. 2007
Abstract We further present some semi-discrete modifications to the cubically convergent iterative methods derived by Kanwar and Tomar (Modified families of Newton, Halley and Chebyshev methods, Appl. Math. Comput. http://dx.doi.org/ 10.1016/j.amc.2007.02.119) and derived a number of interesting new classes of third-order multi-point iterative methods free from second derivatives. Furthermore, several functions have been tested and all the methods considered are found to be effective and compared to the well-known existing third and fourth-order multi-point iterative methods. Keywords Newton’s method . Halley’s method . Chebyshev’s method . One-point iterative method . Multi-point iterative method . Traub–Ostrowski’s method AMS 65H05
1 Introduction Consider a nonlinear equation f ð xÞ ¼ 0;
ð1:1Þ
where f(x) is a real valued function. Let f(x)∈C2[a,b] for sufficiently small interval [a,b] and let r∈(a,b) be a simple root of (1.1).
V. Kanwar (*) University Institute of Engineering and Technology, Panjab University Chandigarh, Chandigarh 160 014, India e-mail:
[email protected] S. K. Tomar Department of Mathematics, Panjab University Chandigarh, Chandigarh, India e-mail:
[email protected]
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Numer Algor (2007) 44:381–389
Newton’s method is an excellent method for solving nonlinear equation (1.1) when one is near the root and has quadratic convergence. The applications of the Newton’s method and to achieve the convergence, it is wanted that f′(x) ≠ 0 in a neighbourhood of the required root. Halley’s method [1, 2], Chebyshev’s method [1] and Chebyshev–Halley type methods [3] are famous third-order iteration for solving nonlinear equation (1.1) and are close relative of Newton’s method. The Chebyshev– Halley type formula with third-order convergence [3] is written as Tf ð xÞ f ð xÞ 1 ϕð xÞ ¼ x 0 1þ ; ð1:2Þ f ð xÞ 2 1 l Tf ð x Þ where λ is an arbitrary real parameter and Tf ð xÞ ¼
1 f ð xÞf ¶¶ð xÞ : 2 f ¶ 2 ð xÞ
ð1:2aÞ
Chebyshev–Halley type methods have two problems, which restrict their practical applications rigorously. The first problem is that like Newton’s method, these methods require the condition that f′(x)≠0 in the vicinity of the root. The second problem is that these methods require the computation of second order derivative. The proposed methods are aimed to taken care of these. The first problem was taken recently by Kanwar and Tomar [6]. They derived third-order Halley and Chebyshev-type iterative formulae and demonstrated their applications through examples. However, these new methods also need the information of second order derivative during computing process. Constructing iterative methods with cubic or quartically convergence, not requiring the computation of second derivative is quite interesting and important from application point of view. For the second problem, there has been some progress on third-order Newtontype iterative methods that do not require the computation of second order derivative. These multi-point methods for single variable non-linear equations have been studied recently by Weerakoon and Fernando [4], Homeier [8], Frontini and Sormani [7] and Kou et al. [10]. Nedzhibov et al. [5] also derived some new families of multi-point iterative methods by discretizing the second-order derivative of Chebyshev–Halley type family (1.2). These methods calculate the new approximations to a zero of the given function by sampling per iteration the function and possibly its derivatives for a number of values of the independent variables. All these techniques are variants of Newton’s method and the main practical difficulty associated with these multi-point families is that they fail miserably if at any stage of computation, the derivative of the function is either zero or very small in the vicinity of the root. In the present paper, we are concerned with developing and unifying the class of third-order multi-point iterative methods free from second derivative. The main idea of the proposed methods lies in discretization of second-order derivative involved in the modified Chebyshev–Halley type family derived by Kanwar and Tomar [6]. An additional feature of these processes is that they may possess a number of disposable parameters which can be used to ensure the convergence is to a certain order for simple zeros.
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The modified Chebyshev–Halley type methods, derived by Kanwar and Tomar [6] is given by Tf ð xÞ f ð xÞ 1 1 þ ϕ ð xÞ ¼ x 0 ; ð1:3Þ f ð xÞ þ a 0 f ð xÞ 2 1 l T f ð xÞ where Tf ð xÞ ¼
f ð xÞ½f 00 ð xÞ þ 2α0 f 0 ð xÞ ff 0 ð x Þ þ α 0 f ð x Þ g2
;
ð1:3aÞ
α′ is a scaling parameter and λ∈
