On a New Efficient Steffensen-Like Iterative Class by Applying a Suitable Self-Accelerator Parameter

It is attempted to present an efficient and free derivative class of Steffensen-like methods for solving nonlinear equations. To this end, firstly, we construct an optimal eighth-order three-step uniparameter without memory of iterative methods. Then the self-accelerator parameter is estimated using Newton's interpolation in such a way that it improves its convergence order from 8 to 12 without any extra function evaluation. Therefore, its efficiency index is increased from 81/4 to 121/4 which is the main feature of this class. To show applicability of the proposed methods, some numerical illustrations are presented.


Introduction
Kung and Traub are pioneers in constructing optimal general multistep methods without memory. They devised two general -step methods based on interpolation. Moreover, they conjectured any -step methods without memory using + 1 function evaluations may reach the convergence order at most 2 [1]. Accordingly, many authors during the last years, specially the four past years, are attempted to construct iterative methods without memory which support this conjecture with optimal order .
Although construction of optimal methods without memory is still an active field, however, much attention has not been paid for developing methods with memory. Based on our best knowledge, Traub in his book introduces the first method with memory. The main feature of these methods is that they improve convergence order as well as efficiency index without any new function evaluations. Indeed, Traub changed Steffensen's method slightly as follows (see [18, pp. 185-187]): 0 , 0 , 0 are given suitably, , 0 ̸ = ∈ , = 0, 1, 2, . . . , The parameter is called self-accelerator and method (1) has convergence order 2.41. It is still possible to increase the convergence order using better self-accelerator parameter based on better Newton interpolation. Free-derivative can be considered as another virtue of (1).
In this work, motivated by Traub's work (1), we construct a new class of methods with memory. To this end, we first try to devise a new optimal free-derivative three-step without memory of iterative methods with eight order of convergence and using merely four function evaluations per step. In other words, our first step is the same as Traub's method (1). The second and third steps use combination Steffensen-like methods and weight function idea so that we achieve an optimal class of methods without memory. Finally, we apply a self-accelerator parameter to extend it to with memory case. We remember two main properties of this work: increasing 2 The Scientific World Journal efficiency index without any new functional evaluations and nonusing derivatives of a given function.
We use the symbols → , , and ∼ according to the following conventions [18]: if lim → ∞ ( ) = , we write ( ) → or → . If lim → ( ) = , we write ( ) → or → . If / → , where is a nonzero constant, we write = ( ) or ∼ . Let ( ) be a function defined on an interval , where is the smallest interval containing + 1 distinct nodes 1 , 2 , . . . , . The divided difference [ 0 , 1 , . . . , ] with th-order is defined as follows: Moreover, we recall the definition of efficiency index (EI) as = 1/ , where is the order of convergence and is the total number of function evaluations per iteration.
This work is organized as follows: Section 2 present construction and error analysis of optimal three-step class of without memory class. Section 3 is devoted to with memory extension. Numerical results are demonstrated in Section 5. We sum up this work in Section 5.

Derivative-Free Three-Point Method
This section concerns construction a new class of threestep free-derivative methods without memory for solving nonlinear equations. In the next section, it is extended to its with memory cases. To this end, let us first start with the following three-step Steffensen-type [23] initiative: , This scheme is not optimal in the sense of Kung and Traub [1] as it is of fourth-order convergence using four functions evaluations per iteration. In other words, its error equation has the form +1 = (1 + 1 ) Therefore, some modifications based on applying weight function ideas must be considered in such a way that the scheme (3) changes into an optimal method. Accordingly, we put forward the following iterative plan: , The main contribution of this section lies in the following Theorem which provides sufficient conditions for drawing optimal three-step iterations without memory class.
If the initial approximation 0 is sufficiently close to the zero of a function , then the convergence order of the family (5) is eight.
In the same manner, we can see that the coefficient of 6 is To vanish the coefficient of 6 , set 0,1 = 1, 0,1 = 2,0 = 2,1 = 2,0 = 0, and we conclude similarly that As in the above cases, choosing 1,1 = 2, 1,0 = 0, and 3,0 = 2 − 2 (13 + ( ) (7 + ( )))))] Some simple but efficient weight functions satisfying the conditions of Theorem 1 are The Scientific World Journal In the next section we introduce a new three-step method with memory. The efficiency index of the optimal class (5) is = 8 1/4 . we extent proposed class (5) to its with memory version, using an accelerator parameter, which improves the efficiency index to 12 1/4 .

A New Method with Memory
Looking at the error equation (20) of the class (5) reveals that we can increase the convergence order of this class if the crucial element 1 + ( ) vanishes. This can be done if = −1/ ( ). Although this is true theoretically, it is not possible practically since is unknown. Fortunately, during the iterative process (5), finer approximations to are generated by the sequence { }, and therefor we try to obtain a good approximate for ( ). Each iteration, , , , , and +1 , are accessible, except at the initial step. Hence, we can interpolate ( ) using these nodes. It is natural that we estimate the best interpolator, and as a result we consider Newton interpolating polynomial as follows: In the next theorem we prove that if ≃ −1/ 4 ( ), then convergence order of the proposed class in Theorem 1 improves to 12.

Theorem 2.
Suppose that 0 is an approximation to a simple zero of , then the R-order of convergence of the three-point method (5) is at least 12.
Remark 3. If we use lower Newton interpolation, we achieve lower -order.

Numerical Results
In this section, we test our proposed methods and compare their results with some other methods of the same order of convergence. First, we introduce some concrete methods based on the proposed class in this work.
Methods Methods For comparison purposes, we consider the following methods 8 The Scientific World Journal  Table 4: ( ) = exp( 2 + cos( ) − 1) sin( ) + log( sin( ) + 1), 0 = 0.6, = 0, 0 = −0.1. Methods By | − | we denote approximations to the zero , (− ) stands for × 10 − , and the computational order of convergence (COC). Here, COC is defined by [16]: Also the following functions are used: Tables 1 and 2 show numerical results for various optimal without memory methods (31)-(38). It is clear that all these methods behave very well practically and confirm their relevant theories. Tables 3 and 4 present numerical results for various with memory methods (31)-(38). It is also clear that all these methods behave very well practically and confirm their relevant theories. They all provide 12th-order of convergence asymptotically without any new function evaluations.

Conclusions
In this work we proposed a new optimal class of methods without and with memory for computing simple root of a nonlinear equation. Its without and with memory methods attain 8 and 12 orders of convergence, respectively, using only four function evaluations per iterations. This class is freederivative which can be considered as another virtue for it. All together, we managed to increase efficiency index of methods without memory from 8 1/4 to 12 1/4 using a very suitable selfaccelerator parameter based on Newton interpolation.