1. Introduction

In this work, we consider iterative methods to find a simple root of a nonlinear equation

(1)

where is a scalar function on the open interval .

The constructing of higher-order convergence methods has attracted a lot of attention from both theoretical as well as practical point of view, see for example [1-9] and references therein.

King in [3] developed a one-parameter family of fourth-order methods, which is written as

(2)

where is a parameter. In particular, the famous Ostrowski’s method is a member of this family when

Recently, based on King’s or Ostrowski’s method, some optimal order iterative methods have been proposed and analyzed for solving nonlinear equations [1, 2, 5, 6, 8, 9]. In particular, Bi et al [1] presented a new family of eighth-order methods:

(3)

where and represents a real-valued function, satisfying the conditions and and divided differences are denoted by

Wang and Liu developed in [9] a new modified Ostrowski’s method:

(4)

Another modifications of Ostrowski’s method were developed in [2, 5, 6]. The first two steps are the same as in (4) and the third step is different in these methods.

All the above mentioned optimal eighth-order methods have the efficiency index In this paper based on King’s method (2), we derive a new family of eighth-order methods for any It has a form:

(5)

which includes the method (4) proposed in [9] as a particular case when It requires, as in (3), (4) the evaluations of only three function and one first-order derivative per iteration. Therefore it has also efficiency index

2. Construction of a Family of Iterative Methods and Convergence Analysis

We consider the following iterative methods

(6)

(7)

where is a real parameter to be determined.

Note that the two first steps in (6) and (7) is King’s iteration and the third step is a linear combination of the two approximations and obtained by preceding steps. We call this iteration (6) and (7) as a modified King’s iterations.

It is well known that the King’s method prescribed by the two first steps in (6) and (7) has a optimal fourth order convergence [4]. The third step in (6) and (7) can be considered as a accelerating procedure for King’s method.

From (7) it clear that belongs to interval connecting and under condition and does not belong to this interval when Our aim is to find optimal value in (7) such that the new approximation will be situated more close to as compared with and From this we deduce that

(8)

and

(9)

Now we proceed to analyse the convergence of iteration (6) and (7). For convenience, we rewrite as

(10)

We use Taylor expansion of around

(11)

Now we approximate using already computed function values. This can be done by the method of undetermined coefficients, such that

(12)

By using the Taylor expansions of around we obtain the following linear system of equations

(13)

which has a unique solution:

(14)

where

(15)

A simple calculations give us

(16)

Substituting (16) into (12) we get

(17)

where

(18)

Substituting (17) into (11) we get

(19)

Now we are ready to state the following convergence theorem for the family of methods (6) and (7).

Theorem 1 Assume that the function is sufficiently differentiable and has a simple zero If the initial point is sufficiently close to and the parameter is chosen by

(20)

Then the method defined by (6) and (7) converges to with eighth-order.

Proof. The first term in brackets in (19) vanishes under choice (20). From (6) we see that

(21)

Since the King’s method has a fourth order of convergence, we also have

(22)

If we take (18) and (21), (22) into account, then from (20) we get

(23)

Using (21), (22) and (23) in (19) we arrive

It means that there exists such that

(24)

On the other hand, by mean-value theorem we have

(25)

where point lying between and

Since on the interval there exist the constants and such that

(26)

Then using (24) and (26) in (25) we obtain

which completes the proof of Theorem 1.

According to (10) and (20) the third step (7) reads as

(27)

It is easy to show that

(28)

Then the proposed iterative methods (6) and (7) can be rewritten as (11).

We call the value t given by (20), the optimal one.

Using (6) and (18) in (20) we obtain

(29)

where

By virtue of (21) and (22) we have

Then from (29) we have

(30)

The factor because of is small enough. Thus the inequalities (8) and (9) will be hold for the approximations and sufficiently close to

Table 1.

3. Numerical Experiments

We takes six examples from [9],

All numerical calculations were performed using Maple 18 system. To study the convergence of iterations, we compute the computational order of convergence (COC) of using the formulae

(31)

where are three consecutive approximations of iterations.

In this Table 1, we present the results of some numerical tests to compare the efficiencies of the methods. We employed, the BM8 method with ((4) in [9]), the BM8-2 method with ((5) in [9]), the method (12) in [9], the KT method ((21) in [9]) and new method (6) and (7). In Table 1, are the test functions, is the initial approximation.

The factor in the brackets denotes From tables we see that the COC perfectly coincides with the theoretical order and new method (6) and (7) is comparable with other optimal order methods.

4. Conclusions

We propose a new family of eighth–order methods based on King’s methods. The high order of convergence is obtained by acceleration procedure. Numerical results clearly demonstrate the theoretical analysis (speed of convergence). Moreover, our acceleration procedure can also be applied to any iteration, for which will be devoted forth coming paper.

ACKNOWLEDGMENTS

The work was supported partially by the Foundation of Science and Technology of Mongolia under grant SST_007/2015.