New computationally efficient formula for backward-pass fixed-interval smoother and its UD factorisation algorithm

A UD factorisation-based backward-pass fixed-interval smoother that is numerically reliable and stable is derived for linear stochastic discrete-time systems. A computationally efficient recursion of a classic backward-pass smoother is first obtained, so that the smoother can exclude the well-known shortcomings of the classic version and utilise the outputs of a forward-pass information filter. This recursion formula is then applied to construct the UD smoother using three fundamental UD algorithms. It is shown that, compared with Bierman´s backward-pass UD smoother (1977, 1983), the UD smoother presented can provide an improvement in computation speed and computer storage for time-invariant systems, as well as the forward-pass UD smoother, but cannot avoid the computation of an inversion of the state-transition matrix for time-varying systems.