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 to the equation . We will also assume that admits initial conditions in at a non-singular point .

Given a broken-line path between and , which avoids the singularities of and with vertices in , we have shown in a previous paper how to compute digits of the analytic continuation of along in time . In a second paper, this result was generalized to the case when is allowed to be a regular singularity, in which case we compute the limit of when we approach the singularity along .

In the present paper, we treat the remaining case when the end-point of 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 decimal digits in time . As a consequence, the entries of the Stokes matrices for at each singularity may be approximated with the same time complexity.

Keywords: algorithm, holonomic function, accelero-summation, Stokes matrix

A.M.S. subject classification: 33-04, 30-04, 40-04, 33F05, 33E30, 40G10, 30B40

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February 2021: Errata, Corrected version