Stable computation of the complex roots of unity

In this paper, we show that the problem of computing the complex roots of unity is not as simple as it seems at rst. In particular, the formulas given in a standard programmer's reference book Knuth, Seminumerical Algorithms, 1981] are shown to be numerically unstable , giving unacceptably large error for moderate sized sequences. We give alternative formulas, which we show to be superior both by analysis and experiment.