TITLE: Computing roots of high degree polynomial numerically.
QUESTION [5 upvotes]: Here is my problem ; for my research, I believe that the complex numbers I am looking at are precisely the (very large) set of roots of some high degree polynomial, of degree $\sim 2^n$ where $1 \le n \le 10 \sim 15$. Mathematica has been running for the whole day on my computer just for $2^{10}$ even though $2^9$ took half an hour, and I wondered if any other program out there would be faster than Mathematica so that I could compute more of those roots. If I had more examples to compute it would REALLY help. The thing I need the program to be able to do is simple : I give you a polynomial of very high degree, and I want to numerically plot its complex roots. I don't care about multiplicity.
Thanks in advance,
EDIT: Marvis, here is my code for computing the nested polynomial $p^m(x) - x$, where $p^2(x) = p(p(x))$.

function r = polycomp(p,q);
r = p(1);
for k = 2:length(p);
r = conv(r,q);
r(end) = r(end) + p(k);
end

All I do afterwards is a loop with

r = [1 0]
for i = 1:n
r = polycomp(r,p)
end

where $n$ is my loop length and $p$ is my polynomial.

REPLY [2 votes]: I'm coming to the conversation late, but I note that there is no discussion of the accuracy of the determined roots. Roots are highly sensitive to perturbations in the polynomial coefficients, i.e. numerical precision is a key determinant of root accuracy particularly for huge polynomials.  
My question: MATLAB may be faster but is it more accurate than Mathematica? Did you check to see whether your $2^n$ roots aren't mostly wildly inaccurate?