puuuurty
i'd like to know the error involved, and if it has to do with the fractal patterns seen. finding roots of polynomials is difficult to do exactly. typically iterative methods are used that can only converge to the root.
i'd like to know the error involved, and if it has to do with the fractal patterns seen. finding roots of polynomials is difficult to do exactly. typically iterative methods are used that can only converge to the root.
http://en.wikipedia.org/wiki/Abel-Ruffini_theorem
Nonetheless, the iterative approximation algorithms are incredibly efficient and numerically stable. There's one really clever trick that actually uses floating-point imprecision to do something you couldn't do in exact arithmetic (inverting a degenerate matrix), that converges to within machine-epsilon in about three iterations.
[1] In the general case; special cases may be tractable, e.g. x^n-1=0.
"The theorem says that not all higher-degree equations have solutions which can be expressed by performing a finite number of operations... Some polynomials of arbitrary degree are indeed solvable with a finite number of such operations."