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For the iterative method

$$z_{n+1} = z_n - \frac{f(z_n)}{f'(z_n)}\hspace{2cm}(*)$$

where $f:U \to \mathbb{C}$ and $U$ is an open subset of $\mathbb{C}$. I know that for stationary points ($f'(z)=0$) and points which enter a cycle, the method will not converge. But are there some sufficient features of an initial guess $z_0 \in U$ so the method converges? I don't want to write that (*) "will converge with luck".

Thanks for your help!

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1 Answer 1

up vote 4 down vote accepted

It's complicated. See http://en.wikipedia.org/wiki/Newton_fractal for some idea of just how complicated it can get.

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