Past input reconstruction in fast least-squares algorithms

This paper solves the following problem: given the computed variables in a fast least-squares prediction algorithm, determine all past input sequences that would have given rise to the variables in question. This problem is motivated by the backward consistency approach to numerical stability in this algorithm class; the set of reachable variables in exact arithmetic is known to furnish a stability domain. Our problem is equivalent to a first- and second-order interpolation problem introduced by Mullis and Roberts (1976) and studied by others. Our solution differs in two respects. First, relations to classical interpolation theory are brought out, which allows us to parametrize all solutions. By-products of our formulation are correct necessary and sufficient conditions for the problem to be solvable, in contrast to previous works, whose claimed sufficient conditions are shown to fall short. Second, our solution obtains any valid past input as the impulse response of an appropriately constrained orthogonal filter, whose rotation parameters derive in a direct manner from the computed variables in a fast least-squares prediction algorithm. Formulas showing explicitly the form of all valid past inputs should facilitate the study of what past input perturbation is necessary to account for accumulated arithmetic errors in this algorithm class. This, in turn, is expected to have an impact in studying accuracy aspects in fast least-squares algorithms.