A family of composite fourth-order iterative methods for solving nonlinear equations
Introduction
In this paper, we consider iterative methods to find a simple root α, i.e., f(α) = 0 and f′(α) ≠ 0, of a nonlinear equation f(x) = 0.
Newton’s method is the well-known iterative method for finding α by usingthat converges quadratically in some neighborhood of α.
There exist many iterative methods improving Newton’s method for solving nonlinear equations. However, many of those iterative methods depend on the second or higher derivatives in computing process which make their practical application restricted strictly. As a result, Newton’s method is frequently and alternatively used to solve nonlinear equations because of higher computational efficiency.
In recent years, there has been some progress on iterative methods improving Newton’s method with cubic convergence that do not require the computation of second derivatives for solving nonlinear equations, see [1], [2], [3], [4], [5], [6], [7], [8], [9] and the reference therein. Furthermore, in [10], [11], [12] several iterative fourth-order methods which are free from second derivatives and that do require only three evaluations of both the function and its derivatives are proposed.
Motivated and inspired by the research going on in this area, in this paper, we suggest and analyze a family of new fourth-order iterative methods. The new methods do not require the computation of second or higher derivatives; besides, they require only two evaluations of the function and one of its derivative. We show that the iterative methods so obtained have all order of convergence four, so that they have better efficiency than Newton’s method. Several numerical examples are given to show the efficiency and the performance of the methods presented in this contribution.
Section snippets
Families of iterative methods
In the sequel, whenever we mention that an iteration function ϕ is of order p, it means that the corresponding iterative method defined by xn+1 = ϕ(xn) is of convergence order p, that is, the error ∣α − xn+1∣ is proportional to ∣α − xn∣p as n → ∞. We refer to [13] for further details about the order of an iteration function.
Let ϕ, ζ and η be iteration functions of order three. Now, we consider the function Φ defined in the following linear combination formwhere θi
Numerical examples and conclusions
All computations were done using MAPLE using 64 digit floating point arithmetics (Digits : = 64). We accept an approximate solution rather than the exact root, depending on the precision (ϵ) of the computer. We use the following stopping criteria for computer programs: (i) ∣xn+1 − xn∣ < ϵ, (ii) ∣f(xn+1)∣ < ϵ, and so, when the stopping criterion is satisfied, xn+1 is taken as the exact root α computed. For numerical illustrations in this section we used the fixed stopping criterion ϵ = 10−15.
We present
Conclusion
In this work we presented an approach which can be used to constructing family of fourth-order iterative methods that do not require the computation of second or higher derivatives; besides the resulting methods do require only two evaluations of the function and one of its first derivative. Some of the obtained methods were also compared in their performance and efficiency to various other iteration methods of the same order, and it was observed that they demonstrate at least equal behavior.
References (15)
- et al.
A variant of Newton’s method with accelerated third-order convergence
Appl. Math. Lett.
(2000) A modified Newton method with cubic convergence
J. Comput. Appl. Math.
(2003)A composite third order Newton–Steffensen method for solving nonlinear equations
Appl. Math. Comput.
(2005)On Newton-type methods with cubic convergence
J. Comput. Appl. Math.
(2005)Iterative methods improving Newton’s method by the decomposition method
Comput. Math. Appl.
(2005)- et al.
Nondiscrete induction and iterative processes
(1984) - et al.
Some variants of Newton’s method with third-order convergence
J. Comput. Appl. Math.
(2004)
Cited by (33)
Third order derivative free SPH iterative method for solving nonlinear systems
2015, Applied Mathematics and ComputationIterative methods for solving nonlinear equations with finitely many roots in an interval
2012, Journal of Computational and Applied MathematicsUnifying fourth-order family of iterative methods
2011, Applied Mathematics LettersA family of fourth-order Steffensen-type methods with the applications on solving nonlinear ODEs
2011, Applied Mathematics and ComputationZero-finder methods derived using Runge-Kutta techniques
2011, Applied Mathematics and ComputationCitation Excerpt :No doubt that Newton’s method plays a central role for solving nonlinear equations due to its order and efficiency. Variations and improvements of this method with an additional evaluation of the function, additional evaluation of the first derivative, or a change in the point of evaluation, have drawn the attention of many researchers and recently some papers in this direction have appeared in the literature of the subject [3–5,9–11,13–15,20,27]. The goal of this paper is to give a technique to obtain zero-finder families of methods based on Runge–Kutta techniques for the numerical solution of differential equations.
Construction of optimal order nonlinear solvers using inverse interpolation
2010, Applied Mathematics and ComputationCitation Excerpt :The first optimal two-point methods were constructed by Ostrowski [1], Jarratt [5,6], King [7] and Kung and Traub [4]. New optimal two-point methods have been developed at the beginning of this century, see, e.g., [8–18]. Kung and Traub [4] and Sharma and Goyal [19] have developed the fourth order two-step optimal methods that do not require any derivative.