A Look-Ahead Block Schur Algorithm for Toeplitz-Like Matrices

Abstract

We derive a look-ahead recursive algorithm for the block triangular factorization of Toeplitz-like matrices. The derivation is based on combining the block Schur/Gauss reduction procedure with displacement structure and leads to an efficient block-Schur complementation algorithm. For an $n \times n$ Toeplitz-like matrix, the overall computational complexity of the algorithm is $O( rn^2 + \frac{n^3 }{t} )$ operations, where r is the matrix displacement rank and t is the number of diagonal blocks. These blocks can be of any desirable size. They may, for example, correspond to the smallest nonsingular leading submatrices or, alternatively, to numerically well-conditioned blocks.