On Solving Block Toeplitz Systems Using a Block Schur Algorithm

This paper presents a block Schur algorithm to obtain a factorization of a symmetric block Toeplitz matrix. It is inspired by the various block Schur algorithms that have appeared in the literature but which have not considered the innuence of performance tradeoos on implementation choices. We develop a version based on block hyperbolic Householder reeectors by adapting the representation schemes for block Householder reeectors in the literature to the hyperbolic case. The basic algorithm is applicable to symmetric positive deenite Toeplitz matrices. Leading evidence is presented that, under certain circumstances, performance gains can be obtained by foregoing some of the Toeplitz structure by using have a block size larger than the actual block size given by the structure of the matrix. This allows the block algorithm to also be used to factor eeciently standard symmetric positive deenite Toeplitz matrices. An extension to the algorithm that can be used to solve symmetric Toeplitz systems that are indeenite is also presented. If a singular principal submatrix is encountered during the factorization, the matrix is perturbed and an approximate factorization is obtained. The error introduced into the solution is then reduced to acceptable levels by applying iterative reenement. Typically two steps are suucient.