Finding all complex zeros of a high-degree polynomial Given a large univariate polynomial, say of degree 200 or more, is there a procedural way of finding all the complex roots? By "roots", I mean complex decimal approximations to the roots, though the multiplicity of the root is important. I have access to MAPLE and the closest function I've seen is:
with(RootFinding):
Analytic(Z,x,-(2+2*I)..2+2*I);

but this chokes if Z is of high degree (in fact it fails to complete even if deg(Z)>15). 

I think this might help. 
Rouche's Theorem: Let $D$ be a bounded domain, with piecewise smooth boundary, $\partial{D}$. Let $f(z)$ and $h(z)$ be analytic on $D \cup \partial{D}$. If $|h(z)| < |f(z)$ for $z \in \partial{D}$, then $f(z)$ and $f(z)+h(z)$ have the same number of zero's in $D$, counting multiplicities.
Example: We find the zeros of the function $p(z)=z^{6}+9z^{4}+z^{3}+2z+4$, inside the unit circle. 
For using Rouche's theorem, let $p(z)= f(z) + h(z)$, where $f(z)$ dominates, $h(z)$ inside the unit circle. 
Consider $f(z) = 9z^{4}$, which has four zeros inside the unit circle, all at the origin. $h(z) =z^{6}+z^{3}+2z+4$, which satisfies, $|h(z)| < |f(z)|$ for $|z|=1$. Therefore by Rouche's theorem $p(z)$ also has 4 zeros inside the unit circle. 
