Formally biorthogonal polynomials and a look-ahead Levinson algorithm for general Teoplitz systems

Abstract

Systems of linear equations with Toeplitz coefficient matrices arise in many important applications. The classical Levinson algorithm computes solutions of Toeplitz systems with only O( n 2) arithmetic operations, as compared to O( n 3) operations that are needed for solving general linear systems. However, the Levinson algorithm in its original form requires that all leading principal submatrices be nonsingular. In this paper, an extension of the Levinson algorithm to general Toeplitz systems is presented. The algorithm uses look-ahead to skip over exactly singular as well as ill-conditioned leading submatrices, and at the same time it still fully exploits the Toeplitz structure. In our derivation of this algorithm, we make use of the intimate connection of Toeplitz matrices with formally biorthogonal polynomials. In particular, the occurence of singular or ill-conditioned submatrices corresponds to a breakdown or near-breakdown in the standard recurrence relations for biorthogonal polynomials. We present new general recurrence relations that connect successive pairs in any given subsequence of all existing formally biorthogonal polynomials. These recurrences then immediately lead to the proposed look-ahead Levinson algorithm for solving Toeplitz systems. Implementation details for this algorithm and operation counts are given. Numerical experiments for Toeplitz systems with ill-conditioned submatrices are reported.