France.

lyon.fr In this paper we prove that computing the solution of an initial-value problem y ̇ = p(y) with initial condition y(t0) = y0 ∈ Rd at time t0 + T with precision e−µ where p is a vector of polynomials can be done in time polynomial in the value of T, µ and Y = supt06u6T ‖y(u)‖∞. Contrary to existing results, our algorithm works for any vector of polynomials p over any bounded or unbounded domain and has a guaranteed complexity and precision. In particular we do not assume p to be fixed, nor the solution to lie in a compact domain, nor we assume that p has a Lipschitz constant. 1