On modern array algorithms for optimal discrete filtering

Nowadays, computational methods for optimal estimation have become an independent field of research and have received a great progress. Modern numerically efficient array algorithms are attractive not only because of their robustness to machine round-off errors, but additionally because of utilization various types of matrix orthogonal transformations. Thus, their design pattern is well suited for parallel implementations on modern computing systems. These properties allow to develop new efficient information technologies, in particular, the techniques that are applicable for solving real-time problems as well as for processing big data arrays. This paper gives a brief survey of modern array algorithms for optimal linear discrete-time filtering. Four large classes of array algorithms are considered: square-root array algorithms, array algorithms based on weighted orthogonalization, $J$-orthogonal array algorithms and methods based on singular value decomposition. We suggest a classification of array algorithms according to the type of the utilized matrix orthogonal transformation on the basis of which these algorithms are designed. Such classification suggests a more simple way for understanding the array filtering methods' design and gives a choice for finding their most effective implementation for estimating multivariable discrete-time linear stochastic systems. The computational aspects of array algorithms are investigated. It includes the numerical stability to machine round-off errors, and discussion of efficient software implementation for the array algorithms under examination. Finally, the array algorithms investigated in this paper are algebraically equivalent to the conventional implementation of the discrete-time Kalman filter, but they possess the significantly improved computational properties. The results of the presented comparative study allow to conclude that the use of array algorithms in solving practical problems helps to obtain numerically efficient and reliable solutions.