Newton's method and FFT trading

Let C[[z]] be the ring of power series over an effective ring C. In Brent and Kung (1978), 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 for multiplying 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 on the order r of the equation, which is exponential for the original method (van der Hoeven, 2002) and quadratic for a recent improvement (Bostan et al., 2007). In this paper, we present a technique for further improving the dependency on r, by applying Newton's method up to a lower order, such as n/r, and trading the remaining Newton steps against a lazy or relaxed algorithm in a suitable FFT model. The technique leads to improved asymptotic complexities for several basic operations on formal power series, such as division, exponentiation and the resolution of more general linear and non-linear systems of equations.