A reformulation of the Rauch-Tung-Streibel discrete time fixed interval smoother

A smoothing algorithm was presented by G.J. Bierman (1983) which is much more efficient and numerically reliable than the well-known R-T-S (Runch-Tung-Streibel) smoother. The Bierman-formulated smoother is extended to account for situations with singular estimate error covariances that arise in applications. The new algorithm, based on numerically stable Givens reflections, is designed to integrate with the UDU^T factorized form of the Kalman filter. Bierman smoother gains that relate to the R-T-S smoother gains, are introduced; the formulation is such that ill-conditioned and possible singular matrices are avoided using a matrix factorization formulation. The new U-D factorized covariance algorithm, based on rank 2 U-D factor modifications, is arranged so that all the positive numerical aspects described in G.J. Bierman (1983) are retained and further enhanced