A family of composite fourth-order iterative methods for solving nonlinear equations

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Abstract

In this paper, we present a family of new fourth-order iterative methods for solving nonlinear equations. Per iteration the methods consisting of the family require only two evaluations of the function and one evaluation of its derivative. Several numerical examples are given to illustrate the efficiency and performance of some of the presented methods.

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 usingxn+1=xn-f(xn)f(xn),that 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  xnp 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 formΦ(x)=x-θ1[x-ϕ(x)]-θ2[x-ζ(x)]-θ3[x-η(x)],where 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.

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