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@ cKleinhuis:
To save storage space, I think it would be better to compute the root when the user asks for it. Given a region, the computations, to get the smallest period root inside it, would take some milliseconds, even with big precision and a java implementation. Claude's program is a good candidate for that purpose but there are cases where it fails (encoutering a zero derivative).

Here is attached a new version. I used my own implementation of complex class. Now using long double is only 3 times slower than using doubles. order 14 is computed in about 0.6s instead of 2.62s on an i7 (both multithreaded).
Moreover the difference in speed between using -O3 and -Ofast is small now.