Efficient accelero-summation of holonomic functions

Let be a linear differential operator, where is the field of algebraic numbers. A holonomic function over is a solution f to the equation Lf=0. We will also assume that f admits initial conditions in at a non-singular point . Given a broken-line path γ=z z′ between z and z′, which avoids the singularities of L and with vertices in , 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(nlog3nloglogn). 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 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(nlog4nloglogn). As a consequence, the entries of the Stokes matrices for L at each singularity may be approximated with the same time complexity.