Super-irreducible form of linear differential systems
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The notion of irreducible forms of systems of linear differential equations with formal power series coefficients as defined by Moser [Moser, J., 1960. The order of a singularity in Fuchs' theory. Math. Z. 379-398] and its generalisation, the super-irreducible forms introduced in Hilali and Wazner [Hilali, A., Wazner, A., 1987. Formes super-irreductibles des systemes differentiels lineaires. Numer. Math. 50, 429-449], are important concepts in the context of the symbolic resolution of systems of linear differential equations [Barkatou, M., 1997. An algorithm to compute the exponential part of a formal fundamental matrix solution of a linear differential system. Journal of App. Alg. in Eng. Comm. and Comp. 8 (1), 1-23; Pflugel, E., 1998. Resolution symbolique des systemes differentiels lineaires. Ph.D. Thesis, LMC-IMAG; Pflugel, E., 2000. Effective formal reduction of linear differential systems. Appl. Alg. Eng. Comm. Comp., 10 (2) 153-187]. In this paper, we reduce the task of computing a super-irreducible form to that of computing one or several Moser-irreducible forms, using a block-reduction algorithm. This algorithm works on the system directly without converting it to more general types of systems as needed in our previous paper [Barkatou, M., Pflugel, E., 2007. Computing super-irreducible forms of systems of linear differential equations via Moser-reduction: A new approach. In: Proceedings of ISSAC'07. ACM Press, Waterloo, Canada, pp. 1-8]. We perform a cost analysis of our algorithm in order to give the complexity of the super-reduction in terms of the dimension and the Poincare-rank of the input system. We compare our method with previous algorithms and show that, for systems of big size, the direct block-reduction method is more efficient.