Block matrices with L-block-banded inverse: inversion algorithms

Block-banded matrices generalize banded matrices. We study the properties of positive definite full matrices P whose inverses A are L-block-banded. We show that, for such matrices, the blocks in the L-block band of P completely determine P; namely, all blocks of P outside its L-block band are computed from the blocks in the L-block band of P. We derive fast inversion algorithms for P and its inverse A that, when compared to direct inversion, are faster by two orders of magnitude of the linear dimension of the constituent blocks. We apply these inversion algorithms to successfully develop fast approximations to Kalman-Bucy filters in applications with high dimensional states where the direct inversion of the covariance matrix is computationally unfeasible.