JUN-SEOP SONG 6 Therefore, we obtain P ( z ) = ( 1 + n ∑ j =1 W$_{j}$ z − z$_{j}$ ) n ∏ j =1 ( z − z$_{j}$ ) (10) = W$_{i}$ ∏ j = i ( z − z$_{j}$ ) + ( 1 + ∑ j = i W$_{j}$ z − z$_{j}$ ) n ∏ j =1 ( z − z$_{j}$ ) . P $^{′}$( z$_{i}$ ) P ( z$_{i}$ ) = ( ∑ j = i 1 z$_{i}$ − z$_{j}$ ) + 1 + ∑ j = i W$_{j}$ z$_{i}$ − z$_{j}$ W$_{i}$ . W$_{i}$ = P ( z$_{i}$ ) P $^{′}$( z$_{i}$ ) · 1 + ∑ j = i W$_{j}$ z$_{i}$ − z$_{j}$ 1 − P ( z$_{i}$ ) P ′ ( z$_{i}$ ) ∑ j = i 1 z$_{i}$ − z$_{j}$ = P ( z$_{i}$ ) P $^{′}$( z$_{i}$ ) (1 + O$_{M}$ ( e )) . (12) ˆ z$_{i}$ = z$_{i}$ − 2 P ( z$_{i}$ ) P $^{′}$( z$_{i}$ ) + P $^{′}$( z$_{i}$ − W$_{i}$ ) = z$_{i}$ − 2 P ( z$_{i}$ ) P $^{′}$( z$_{i}$ ) + ( P $^{′}$( z$_{i}$ ) − P $^{′′}$( z$_{i}$ ) W$_{i}$ + 1 $_{2}$P $^{′′′}$( z$_{i}$ ) W 2 i + · · · ) = z$_{i}$ − P ( z$_{i}$ ) P $^{′}$( z$_{i}$ ) − 1 $_{2}$P $^{′′}$( z$_{i}$ ) W$_{i}$ + 1 $_{4}$P $^{′′′}$( z$_{i}$ ) W 2 i + · · · . Now we will find the order of Method 1. If the Taylor’s expansion is applied to P $^{′}$( z$_{i}$ − W$_{i}$ ), then we have Therefore, we have Substituting z = z$_{i}$ in (11), we obtain From (10), it follows that P $^{′}$( z ) P ( z ) = ( ∑ j = i 1 z − z$_{j}$ ) + 1 + ∑ j = i W$_{j}$ z − z$_{j}$ − ( z − z$_{i}$ ) ∑ j = i W$_{j}$ ( z − z$_{j}$ ) 2 W$_{i}$ + ( z − z$_{i}$ ) ( 1 + j = i W$_{j}$ z − z$_{j}$ ) . (11)