Row reduction process for matrices of scalar operators: storing the intermediate results of row transformation

It is well known that if the leading matrix of a linear ordinary differential or difference system is nonsingular, then the determinant of this matrix contains some useful information on solutions of the system. We investigate a kind of non-arithmetic complexity of known algorithms for transforming a matrix of scalar operators to an equivalent matrix which has non-singular frontal, or, leading matrix. In the algorithms under consideration, the differentiation in the differential case and the shift in the difference case play a significant role. We give some analysis of the complexity measured as the number of differentiations or, resp., shifts in the worst case. We not only offer estimates of the complexity written using the O-notation, but we also show that some estimates are sharp and can not be improved.