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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.