content_image/305768.jpg content_image/305774.jpg content_image/305775.jpg content_image/305776.jpg
From (8), we obtain 

<img src='content_image/305768.jpg'>

From Lemma 1, we have the following theorem. 

Theorem 1. If the approximate zero x$_{i}$ grounded from Method 1 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. We easily see that the following equation is satisfied. 

<img src='content_image/305774.jpg'>

That is, 

<img src='content_image/305775.jpg'>

If Q ( z ) = P ( z ) − ∏ n j $_{=1}$( z − z$_{j}$ ), then Q ( z ) is a polynomial of order n − 1, and Q ( z$_{i}$ ) = P ( z$_{i}$ ) for all i . Therefore, Q ( z ) is the Lagrange interpolation of points z$_{1}$, z$_{2}$, . . . , z$_{n}$ , and so we have 

<img src='content_image/305776.jpg'>

