Numerically stable fast recursive least squares algorithms for adaptive filtering using interval arithmetic

Fast recursive least squares algorithms such as the fast Kalman algorithm, the FAEST algorithm, and the FTF algorithm perform least squares adaptive filtering with low computational complexity, which is directly proportional to the filter length. Unfortunately, these highly efficient algorithms suffer from severe numerical instability when implemented using either fixed or floating point digital arithmetic. Small numerical errors due to the finite precision of the computations at each iteration of the algorithm are propagated and accumulate. Eventually the algorithm diverges from the correct solution, often very suddenly. A new approach is used to perform stabilisation. Interval arithmetic is used to provide an upper and a lower bound to the solution produced by the adaptive algorithm, allowing for the possible effects of finite precision arithmetic. If the difference between the upper and the lower bounds becomes excessively large, then the fast RLS algorithm may be reinitialised, preventing divergence