Fast computation of power series solutions of systems of differential equations

High precision expansions of power series solutions of differential equations are needed in various branches of computational mathematics, from combinatorics, where the desired power series is a generating function, to control theory and computational number theory. In this talk, we describe new algorithms for computing the first N coefficients of a power series solution of a system of differential equations, using a number of arithmetic operations which is quasi-linear in N. This extends a classical result due to Brent and Kung. Joint work with F. Chyzak, F. Ollivier, B. Salvy, E. Schost, A. Sedoglavic.