On the Complexity of Composition and Generalized Composition of Power Series

Let F (x) = f1x + f2x + · · · be a formal power series over a field ∆. Let F (0)(x) = x and, for q = 1, 2, . . . , define F (q)(x) = F (q−1)(F (x)). The obvious algorithm for computing the first n terms of F (q)(x) is by the composition analogue of repeated squaring. This algorithm has complexity about log2 q times that of a single composition. Brent [1] showed that the factor log2 q can be eliminated in the computation of the first n terms of (F (x))q by a change of representation, using the logarithm and exponential functions. We show here that the factor log2 q can also be eliminated for the composition problem, unless the complexity of composition is quasi-linear. F (q)(x) can often, but not always, be defined for more general q. We give algorithms and complexity bounds for computing the first n terms of F (q)(x) whenever it is defined. We conclude the paper with some open problems. Comments Only the Abstract is given here. The full paper appeared as [3]. For related work, see [2].