We propose a method for computing the regular singular formal solutions of a linear differential system in the neighbourhood of a singular point. This algorithm avoids the use of cyclic vectors and has been implemented † in the computer algebra system Maple.

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The notion of irreducible forms of systems of linear differential equations as defined by Moser [14] and its generalisation, the super-irreducible forms introduced by Hilali/Wazner in [9] are important concepts in the context of the symbolic resolution of systems of linear differential equations [3,15,16]. In this paper, we give a new algorithm for computing, given an arbitrary linear differential system with formal power series coefficients as input, an equivalent system which is super-irreducible. Our algorithm is optimal in the sense that it computes transformation matrices which obtain a maximal reduction of rank in each step of the algorithm. This distinguishes it from the algorithms in [9, 14, 2] and generalises [7].