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Let be a linear differential operator, where is an effective algebraically closed subfield of . It can be shown that the differential Galois group of is generated (as a closed algebraic group) by a finite number of monodromy matrices, Stokes matrices and matrices in local exponential groups. Moreover, there exist fast algorithms for the approximation of the entries of these matrices.

In this paper, we present a numeric-symbolic algorithm for the computation of the closed algebraic subgroup generated by a finite number of invertible matrices. Using the above results, this yields an algorithm for the computation of differential Galois groups, when computing with a sufficient precision.

Even though there is no straightforward way to find a “sufficient
precision” for guaranteeing the correctness of the end-result, it
is often possible to check *a posteriori* whether the end-result
is correct. In particular, we present a non-heuristic algorithm for the
factorization of linear differential operators.

**Occasions:** ANR Gecko meeting, Toulouse, 2006

**Documents:** slideshow, TeXmacs
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