Newton ’ s method and FFT trading ∗ Joris

Let C[[z]] be the ring of power series over an effective ring C. In [BK78], it was first shown that differential equations over C[[z]] may be solved in an asymptotically efficient way using Newton’s method. More precisely, if M(n) denotes the complexity in order two polynomials of degree < n over C, then the first n coefficients of the solution can be computed in time O(M(n)). However, this complexity does not take into account the dependency of on the order r of the equation, which is exponential for the original method [vdH02] and linear for a recent improvement [BCO+06]. In this paper, we present a technique to get rid of this constant factor, by applying Newton’s method up to an order like n/r and trading the remaining Newton steps against a lazy or relaxed algorithm in a suitable FFT model.