content_image/302208.jpg content_image/302209.jpg content_image/302210.jpg content_image/302211.jpg
From (13), we have the following theorem. 

Theorem 2. If the approximate zero x$_{i}$ grounded from Method 2 is close enough to r$_{i}$ and the order of e$_{i}$ is the same, then the order of ˆ e$_{i}$ is identical, and | ˆ e$_{i}$ | = O$_{M}$ ( | e | $^{3}$) is formed. 

Proof. Since Petkovi´ c’s derivative-free method (6) is of cubic order 

<img src='content_image/302208.jpg'>

(see [6]). Using (14) and the Taylor’s expansion, ˆ z$_{i}$ is calculated as follows. (In this case, 

<img src='content_image/302209.jpg'>

Therefore, the order of ˆ e$_{i}$ is calculated as follows. 

<img src='content_image/302210.jpg'>

Now we apply the derivative-free method to the x ∗ in the midpoint Newton’s method (4) and construct the iteration as follows: 

<img src='content_image/302211.jpg'>

