1. Apr 22, 2010

### math8

Let $$x_{m}$$ and $$x_{m+1}$$ be two successive iterates when Newton's method is applied to a polynomial $$p(z)$$ of degree n. We prove that there is a zero of $$p(z)$$ in the disk

$${z \in \textbf{C}: |z - x_{m}| \leq n|x_{m+1} - x_{m}| }$$.

I suppose we may use $$p'(z)/ p(z) = \sum ^{n}_{j=1} 1/ \left( z-r_{j} \right)$$ where $$r_{1}, r_{2}, \cdots , r_{n}$$ are roots of p.

We need to show there is a root $$\alpha$$ of $$p(z)$$ for which $$|\alpha - x_{m}| \leq n|x_{m+1} - x_{m}|}$$ i.e. for which $$\frac{n}{|\alpha - x_{m}|} \geq \left| \frac{p'(x_{m})}{p(x_{m})} \right| = \sum ^{n}_{j=1} 1/ |x_{m}-r_{j} |$$

But I am not sure how to proceed from there.