A minimal realization algorithm for matrix sequences

We give an algorithm for solving the Padé approximation problem for matrix sequences over an arbitrary field. The algorithm is a multivariate version of one first proposed by Berlekamp and Massey in a coding theory context, the extension being obtained using matrix-fraction descriptions of multivariable systems. The algorithm is recursive and seems to have some computational advantages. Furthermore, our results are in a form that permits easy determination of state-space models from the transfer functions, solving what is called the partial realization problem. Our algorithm also shows how to obtain a characterization of the invariants of this problem.