Fast composition of numeric power series ∗

Let f and g be two convergent power series in R[[z]] or C[[z]], whose first n terms are given numerically with a λ n-bit precision for a fixed constant λ > 0. Assuming that g0=0, we will show in this paper that the first n coefficients of f ◦ g can be computed with a λn-bit precision in time Õ(n). Using Newton iteration, a similar complexity bound holds for power series reversion of g. Our method relies on fast multi-point evaluation, which will be recalled and further detailed for numeric polynomials. We also discuss relaxed variants of our algorithm.