On the Complexity of Composition and Generalized Composition of Power Series

Let F(x) = f1x + f2(x)(x) + . . . be a formal power series over a field Delta. Let F superscript 0(x) = x and for q = 1,2 . . . , define F superscript q(x) = F superscript (q-1) (F(x)). The obvious algorithm for computing the first n terms of F superscript q(x) is by the composition position analogue of repeated squaring. This algorithm has complexity about log 2 q times that of a single composition. The factor log 2 q can be eliminated in the computation of the first n terms of (F(x)) to the q power by a change of representation, using the logarithm and exponential functions. We show the factor log 2 q can also be eliminated for the composition problem. F superscript 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 superscript q(x) whenever it is defined