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We present a practical implementation based on Newton's method to find allroots of several families of complex polynomials of degrees exceeding onebillion ($10^9$) so that the observed complexity to find all roots is between$O(d\ln d)$ and $O(d\ln^3 d)$ (measuring complexity in terms of number ofNewton iterations or computing time). All computations were performedsuccessfully on standard desktop computers built between 2007 and 2012.

journal of computational and applied mathematics

Newton's method in practice II: The iterated refinement Newton method  and near-optimal complexity for finding all roots of some polynomials of very  large degrees