Efficient accelero-summation of holonomic functions

Abstract

Let L ∈ K ( z ) [ ∂ ] be a linear differential operator, where K is the field of algebraic numbers. A holonomic function over K is a solution f to the equation L f = 0 . We will also assume that f admits initial conditions in K at a non-singular point z ∈ K . Given a broken-line path γ = z ⇝ z ′ between z and z ′ , which avoids the singularities of L and with vertices in K , we have shown in a previous paper [van der Hoeven, J., 1999. Fast evaluation of holonomic functions. Theoret. Comput. Sci. 210, 199–215] how to compute n digits of the analytic continuation of f along γ in time O ( n log 3 n log log n ) . In a second paper [van der Hoeven, J., 2001b. Fast evaluation of holonomic functions near and in singularities. J. Symbolic Comput. 31, 717–743], this result was generalized to the case when z ′ is allowed to be a regular singularity, in which case we compute the limit of f when we approach the singularity along γ . In the present paper, we treat the remaining case when the end-point of γ is an irregular singularity. In fact, we will solve the more general problem to compute “singular transition matrices” between non-standard points above a singularity and regular points in K near the singularity. These non-standard points correspond to the choice of “non-singular directions” in Écalle’s accelero-summation process. We will show that the entries of the singular transition matrices may be approximated up to n decimal digits in time O ( n log 4 n log log n ) . As a consequence, the entries of the Stokes matrices for L at each singularity may be approximated with the same time complexity.