content_image/322301.jpg content_image/322302.jpg content_image/322303.jpg content_image/322304.jpg content_image/322305.jpg
By (9), (12) and Lemma 1, we have that 

<img src='content_image/322301.jpg'>

The Chebyshev’s method is defined by 

<img src='content_image/322302.jpg'>

and of cubic order (see [7, Section 5.2]). According to the Chebyshev’s method, we see that 

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

<img src='content_image/322303.jpg'>

<img src='content_image/322304.jpg'>

## III. Modified Newton’s methods combined with Derivative-free method 

In this section, we present two modified Newton’s methods combined with the derivative- free method (6) for finding all simple zeros of a polynomials simultaneously. The one is a form that the derivative-free method is applied to the x ∗ in the trapezoidal Newton’s method (3) as follows: 

which is called the Derivative-free trapezoidal Newton’s method , or simply, Method 2 . 

<img src='content_image/322305.jpg'>

