A look-ahead Schur Algorithm

The classical Schur algorithm computes the LDL T factorization of a symmetric Toeplitz matrix in O(n 2) operations, but requires that all the principal minors of the matrix be nonsingular. Look-ahead schemes have been proposed to deal with matrices that have exactly singular principal minors 9], 11]. Unfortunately, these algorithms cannot be extended to matrices that have ill-conditioned principal minors. Consequently, the relative errors obtained using the classical Schur algorithm on matrices having ill-conditioned principal minors is very poor. In this paper, we propose a look-ahead scheme for such matrices and present empirical results which demonstrate the improvement over the classical Schur algorithm for symmetric Toeplitz matrices with ill-conditioned principal minors.