Backward consistency concept and round-off error propagation dynamics in recursive least-squares algorithms

Abstract

A new approach for the analysis of the propagation of roundoff errors in recursive algorithms is presented. This approach is based on the concept of backward consistency. In general, this concept leads to a decomposition of the state space of the algorithm, and in fact, to a manifold. This manifold is the set of state values that are backward consistent. Perturbations within the manifold can be interpreted as resulting from perturbations on the input data. Hence, the error propagation on the manifold corresponds exactly (without averaging or even linearization) to the propagation of the effect of a perturbation of the input data at some point in time on the state of the algorithm at future times. We apply these ideas to the Kalman filter and its various derivatives. In particular, we consider the conventional Kalman filter, some minor variations of it, and its square-root forms. Next we consider the Chandrasekhar equations, which apply to time-invariant state-space models. Recursive least-squares (RLS) parameter estimation is a special case of Kalman filtering, and hence the previous results also apply to the RLS algorithms. We also consider in detail two groups of fast RLS algorithms: the fast transversal filter algorithms and the fast lattice/fast QR RLS algorithms.