Details:
Title  Efficient accelerosummation of holonomic functions  Author(s)  Joris van der Hoeven  Type  Article in Journal  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 nonsingular point z ∈ K . Given a brokenline 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 endpoint of γ is an irregular singularity. In fact, we will solve the more general problem to compute “singular transition matrices” between nonstandard points above a singularity and regular points in K near the singularity. These nonstandard points correspond to the choice of “nonsingular directions” in Écalle’s accelerosummation 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.  Keywords  Algorithm, Holonomic function, Accelerosummation, Stokes matrix  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S074771710700003X 
Language  English  Journal  Journal of Symbolic Computation  Volume  42  Number  4  Pages  389  428  Year  2007  Edition  0  Translation 
No  Refereed 
No 
