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Efficient accelero-summation of holonomic functions. (English) Zbl 1125.34072
Summary: Let \(L\in\mathbb K(z)[\partial]\) be a linear differential operator, where \(\mathbb K\) is the field of algebraic numbers. A holonomic function over \(\mathbb K\) is a solution \(f\) to the equation \(Lf=0\). We also assume that \(f\) admits initial conditions in \(\mathbb K\) at a non-singular point \(z\in\mathbb K\).
Given a broken-line path \(\gamma=z\rightsquigarrow z'\) between \(z\) and \(z'\), which avoids the singularities of \(L\) and with vertices in \(\mathbb K\), we have shown in a previous paper [J. van der Hoeven, Theor. Comput. Sci. 210, No. 1, 199–215 (1999; Zbl 0912.68081)] how to compute \(n\) digits of the analytic continuation of \(f\) along \(\gamma\) in time \(O(n\log^3n\log\log n)\). In a second paper [J. van der Hoeven, J. Symb. Comput. 31, No. 6, 717–743 (2001; Zbl 0982.65024)], 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 \(\gamma\).
In the present paper, we treat the remaining case when the end-point of \(\gamma\) is an irregular singularity. In fact, we solve the more general problem to compute “singular transition matrices” between non-standard points above a singularity and regular points in \(\mathbb K\) near the singularity. These non-standard points correspond to the choice of “non-singular directions” in Écalle’s accelero-summation process.
We show that the entries of the singular transition matrices may be approximated up to \(n\) decimal digits in time \(O(n\log^4n\log\log n)\). As a consequence, the entries of the Stokes matrices for \(L\) at each singularity may be approximated with the same time complexity.

34M25 Formal solutions and transform techniques for ordinary differential equations in the complex domain
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
34-04 Software, source code, etc. for problems pertaining to ordinary differential equations
Mmxlib; Mathemagix
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