Normalized doubling algorithms for finite shift-rank processes

Recently, various fast "doubling" procedures have been sketched or developed for the inversion of matrices with low shift-rank, e.g., Toeplitz matrices. The subclass of symmetric positive definite matrices is of particular interest in linear estimation, these matrices having the interpretation of covariances of finite shift-rank processes. This paper describes a doubling procedure for such covariance matrices. The procedure evaluates in O(n log2n) operations, both the inverse of an order n covariance and an associated set of parameters of great importance in linear filtering (the reflection coefficients if the covariance is Toeplitz).