On simultaneous row and column reduction of higher-order linear differential systems

In this paper, we define simultaneously row and column reduced forms of higher-order linear differential systems with power series coefficients and give two algorithms, along with their complexities, for their computation. We show how the simultaneously row and column reduced form can be used to transform a given higher-order input system into a first-order system. Finally, we show that the algorithm can be used to compute Two-Sided Block Popov forms as given in Barkatou et al. (2010), . These results extend the previous work in Barkatou et al. (2010), , on second-order systems, and Harris et al. (1968), on first-order systems, to systems of arbitrary order.