Efficient accelero-summation of holonomic functions

• Authors: Joris van der Hoeven
• Venue: Journal of Symbolic Computation
• Year: 2007

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 Lf=0. We will also assume that f admits initial conditions in K at a non-singular point z@?K. Given a broken-line path @c=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 @c in time O(nlog^3nloglogn). 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 @c. In the present paper, we treat the remaining case when the end-point of @c 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 Ecalle'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(nlog^4nloglogn). As a consequence, the entries of the Stokes matrices for L at each singularity may be approximated with the same time complexity.