Inversion of block matrices with block banded inverses: application to Kalman-Bucy filtering

We investigate the properties of block matrices with block banded inverses to derive efficient matrix inversion algorithms for such matrices. In particular, we derive the following: (1) a recursive algorithm to invert a full matrix whose inverse is structured as a block tridiagonal matrix; (2) a recursive algorithm to compute the inverse of a structured block tridiagonal matrix. These algorithms are exact. They reduce the computational complexity respectively by two and one orders of magnitude over the direct inversion of the associated matrices. We apply these algorithms to develop a computationally efficient approximate implementation of the Kalman-Bucy filter (KBf) that we refer to as the local KBf. The computational effort of the local KBf is reduced by a factor of I² over the exact KBf while exhibiting near-optimal performance