On the Minimal Realizations of a Finite Sequence

We develop a theory of minimal realizations of a finite sequence over an integral domain R, from first principles. Our notion of a minimal realization is closely related to that of a linear recurring sequence and of a partial realization (as in Mathematical Systems Theory). From this theory, we derive Algorithm MR. which computes a minimal realization of a sequence of L elements using at most L(5L + 1)/2 R-multiplications. We also characterize all minimal realizations of a given sequence in terms of the computed minimal realization. This algorithm computes the linear complexity of an R sequence, solves non-singular linear systems over R (extending Wiedemannʹs method), computes the minimal polynomial of an R-matrix, transfer/growth functions and symbolic Padé approximations. There are also a number of applications to Coding Theory. We thus provide a common framework for solving some well-known problems in Systems Theory, Symbolic/Algebraic Computation and Coding Theory.