content_image/312772.jpg content_image/312773.jpg content_image/312774.jpg
two modified Newton’s methods combined with the derivative-free method. They are constructed by applying the derivative-free method to the x ∗ in the trapezoidal Newton’s method and the midpoint Newton’s method, respectively. In Section 4, we give a numer- ical comparison between various simultaneous methods for finding zeros of a polynomial. Finally, we conclude that the convergence of all new constructed methods in this paper are similar or superior than other iterative methods of cubic order. 

## II. Weierstrass-like Trapezoidal Newton’s method 

In this section, we construct a new method for finding simultaneously all simple zeros of polynomials of cubic order. By applying the Weierstrass method (5) to the x ∗ in the trapezoidal Newton’s method (3), we derive a new method constructed as follows: 

<img src='content_image/312772.jpg'>

We call (7) the Weierstrass-like trapezoidal Newton’s method , and from this, simply, call it Method 1 . 

The calculation and discussion of the order of Method 1 are similar to those of the Newton-Weierstrass method, which is an alteration of Petkovi´ c’s midpoint Newton’s method (see [5]). From (7), we have the following theorem. 

Lemma 1. For a polynomial P ( z ) , we have 

Proof. By the Taylor’s expansion around r$_{i}$ , we have that 

<img src='content_image/312773.jpg'>

<img src='content_image/312774.jpg'>

