Steffensen-Type Method of Super Third-Order C onvergence for S olving N onlinear E quations

In this paper, a one-step Steffensen-type method with super-cubic convergence for solving nonlinear equations is suggested. The convergence order 3.383 is proved theoretically and demonstrated numerically. This super-cubic convergence is obtained by self-accelerating second-order Steffensen’s method twice with memory, but without any new function evaluations. The proposed method is very efficient and convenient, since it is still a derivative-free two-point method. Its theoretical results and high computational efficiency is confirmed by Numerical examples.


Introduction
Finding the root of a nonlinear equation is a classical problem.It is well-known in scientific computation that Newton's method (NM, see [1]): ( ) ( ) is widely used for root-finding, where 0 x is an initial guess of the root.However, when the derivative f ′ is unavailable or is expensive to be obtained, the derivative-free method is necessary.If the derivative ( ) n f x ′ is replaced by the divided difference in (2), Steffensen's method (SM, see [1]) is obtained as follows: NM/SM converges quadratically and requires two function evaluations per iteration.The efficiency index of them is 2 1.414 = .Besides H.T. Kung and J.F. Traub conjectured that an iterative method based on m evaluations per iteration without memory would arrive at the optimal convergence of order 1 2 m− (see [2]), Traub proposed a self-accelerating two-point method of order 2.414 with memory (see [3]): where ( )  , etc.A lot of self-accelerating Steffensen-type methods were derived in the literature (see [1]- [7]).Steffensen-type methods and their applications in the solution of nonlinear systems and nonlinear differential equations were discussed in [1] [4] [5] [8].Recently, by a new self-accelerating technique based on the second-order Newtonian interpolatory polynomial and M.S. Petkovića proposed a cubically convergent Steffensen-like method (see [7]): In this study, a one-step Steffensen-type method is proposed by doubly-self-accelerating in Section 2, its super-cubic convergence is proved in Section 3, and numerical examples are demonstrated in Section 4.

The Method of Steffensen-Type
By the first-order Newtonian interpolatory polynomial , , should be better than ( ) Therefore, we suggest , e., a two-parameter Steffensen's method: where ( ) and { } n µ are bounded constant sequences.The error equation of ( 7) is . By defining 0 0 recursively as the iteration proceeds without any new evaluation to vanish the asymptotic convergence constant, we establish a self-accelerating Steffensen's method with super quadratic convergence as follows: Furthermore, we propose a one-step Steffensen-type method with super cubic convergence by doubly-selfaccelerating as follows: , ,
) ( ) ) Then, the proof can be completed.for (9), we obtain the same system of two equations: , we prove that the convergence of (9) is of order 3.383.

Numerical Examples
Related one-step methods only using two function evaluations per iteration are showed in the following numerical examples.The proposed method is a derivative-free two-point method with high computational efficiency.Example 1.The numerical results of NM, SM, (4), ( 5) and (9) in Table 1 agree with the theoretical analysis.The computational order of convergence is defined by  4), ( 5) and (9) are in Table 2 for the following nonlinear functions: By Taylor formula, we have

Example 2 .
The numerical results of NM, SM, ( ,
i f x i =