7 NEW SIMULTANEOUS METHODS FOR FINDING ALL ZEROS OF A POLYNOMIAL By (9), (12) and Lemma 1, we have that ˆ z$_{i}$ = z$_{i}$ − P ( z$_{i}$ ) P $^{′}$( z$_{i}$ ) ( 1 − P ′′ ( z$_{i}$ ) 2 P ′ ( z$_{i}$ $_{)}$W$_{i}$ + O$_{M}$ ( e $^{2}$) ) = z$_{i}$ − P ( z$_{i}$ ) P $^{′}$( z$_{i}$ ) · ( 1 + P $^{′′}$( z$_{i}$ ) 2 P $^{′}$( z$_{i}$ ) W$_{i}$ + O$_{M}$ ( e $^{2}$) ) = z$_{i}$ − P ( z$_{i}$ ) P $^{′}$( z$_{i}$ ) · 1 + P $^{′′}$( z$_{i}$ ) P ( z$_{i}$ ) 2 P $^{′}$( z$_{i}$ ) 2 (1 + O$_{M}$ ( e )) + O$_{M}$ ( e $^{2}$) ) = z$_{i}$ − P ( z$_{i}$ ) P $^{′}$( z$_{i}$ ) − P $^{′′}$( z$_{i}$ ) P ( z$_{i}$ ) 2 2 P $^{′}$( z$_{i}$ ) 3 − P ( z$_{i}$ ) P $^{′}$( z$_{i}$ ) ( P $^{′′}$( z$_{i}$ ) P ( z$_{i}$ ) 2 P $^{′}$( z$_{i}$ ) 2 O$_{M}$ ( e ) + O$_{M}$ ( e $^{2}$) ) = z$_{i}$ − P ( z$_{i}$ ) P $^{′}$( z$_{i}$ ) − P $^{′′}$( z$_{i}$ ) P ( z$_{i}$ ) 2 2 P $^{′}$( z$_{i}$ ) 3 − O$_{M}$ ( e $^{3}$) . ˆ x = x − ( 1 + f $^{′′}$( x ) f ( x ) 2 f $^{′}$( x ) 2 ) f ( x ) f $^{′}$( x ) , z$_{i}$ − P ( z$_{i}$ ) P $^{′}$( z$_{i}$ ) − P $^{′′}$( z$_{i}$ ) P ( z$_{i}$ ) 2 2 P $^{′}$( z$_{i}$ ) 3 − r$_{i}$ = O$_{M}$ ( e $^{3}$) . | ˆ e$_{i}$ | = | ˆ z$_{i}$ − r$_{i}$ | = ∣ ∣$^{z$_{i}$}$ − P ( z$_{i}$ ) P $^{′}$( z$_{i}$ ) − P $^{′′}$( z$_{i}$ ) P ( z$_{i}$ ) 2 2 P $^{′}$( z$_{i}$ ) 3 − r$_{i}$ + O$_{M}$ ( e $^{3}$) ∣ = O$_{M}$ ( | e | $^{3}$) . □ ˆ z$_{i}$ = z$_{i}$ − 2 P ( z$_{i}$ ) P $^{′}$( z$_{i}$ ) + P ′ ( z$_{i}$ − W$_{i}$ 1 − P ( z$_{i}$ − W$_{i}$ ) /P ( z$_{i}$ ) ) , (13) The Chebyshev’s method is defined by 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: 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: III. Modified Newton’s methods combined with Derivative-free method which is called the Derivative-free trapezoidal Newton’s method , or simply, Method 2 .