Maehly's Procedure

A method for finding Roots which defines

$\begin{displaymath} P_j(x)={P(x)\over (x-x_1)\cdots(x-x_j)}, \end{displaymath}$

(1)

so the derivative is

$\begin{displaymath} P_j'(x) = {P'(x)\over (x-x_1)\cdots(x-x_j)}-{P(x)\over (x-x_1)\cdots(x-x_j)} \sum_{i=1}^j (x-x_i)^{-1}. \end{displaymath}$

(2)

One step of Newton's Method can then be written as

$\begin{displaymath} x_{k+1} = x_k -{P(x_k)\over P'(x_k)-P(x_k)\sum_{i=1}^j (x_k-x_i)^{-1}}. \end{displaymath}$

(3)