8 JUN-SEOP SONG 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 (see [6]). Using (14) and the Taylor’s expansion, ˆ z$_{i}$ is calculated as follows. (In this case, Therefore, the order of ˆ e$_{i}$ is calculated as follows. Now we apply the derivative-free method to the x ∗ in the midpoint Newton’s method (4) and construct the iteration as follows: z$_{i}$ − W$_{i}$ 1 − P ( z$_{i}$ − W$_{i}$ ) /P ( z$_{i}$ ) − r$_{i}$ = O$_{M}$ ( e $^{3}$) , (14) C$_{j}$ = 1 j ! · P ( j $^{)}$( r$_{i}$ ) P ′ ( r$_{i}$ ) ). 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$_{i}$ ) ) = z$_{i}$ − 2 P r$_{i}$ + e$_{i}$ ) P $^{′}$( r$_{i}$ + e$_{i}$ ) + P $^{′}$( r$_{i}$ + O$_{M}$ ( e $^{3}$)) = z$_{i}$ − 2 ( P ( r$_{i}$ ) + P $^{′}$( r$_{i}$ ) e$_{i}$ + 1 $_{2}$P $^{′′}$( r$_{i}$ ) e 2 i + 1 $_{6}$P $^{′′′}$( r$_{i}$ ) e 3 i + · · · ) ( P $^{′}$( r$_{i}$ ) + P $^{′′}$( r$_{i}$ ) e$_{i}$ + 1 $_{2}$P $^{′′′}$( r$_{i}$ ) e 2 i + · · · ) + ( P $^{′}$( r$_{i}$ ) + P $^{′′}$( r$_{i}$ ) O$_{M}$ ( e $^{3}$) + · · · ) = z$_{i}$ − 2 P $^{′}$( r$_{i}$ ) ( e$_{i}$ + C$_{2}$e 2 i + C$_{3}$e 3 i + O$_{M}$ ( e $^{4}$)) 2 P $^{′}$( r$_{i}$ ) ( 1 + C$_{2}$e$_{i}$ + 3 $_{2}$C$_{3}$ e 2 i + O$_{M}$ ( e $^{3}$) ) = z$_{i}$ − e$_{i}$ ( 1 + C$_{2}$e$_{i}$ + C$_{3}$e 2 i + O$_{M}$ ( e $^{3}$) ) ( 1 − C$_{2}$e$_{i}$ + 3 2 C$_{3}$e 2 i + O$_{M}$ ( e $^{3}$) ) = z$_{i}$ − e$_{i}$ + O$_{M}$ ( e $^{3}$) . | ˆ e$_{i}$ | = | ˆ z$_{i}$ − r$_{i}$ | = ∣ z$_{i}$ − e$_{i}$ + O$_{M}$ ( e $^{3}$) − r$_{i}$ ∣ = O$_{M}$ ( | e | $^{3}$) . □ ˆ z$_{i}$ = z$_{i}$ − P ( z$_{i}$ ) P ′ ( z$_{i}$ − 1 2 · W$_{i}$ 1 − P ( z$_{i}$ − W$_{i}$ ) /P ( z$_{i}$ ) ) , (15)