Fast matrix factorizations via discrete transmission lines

Abstract

Fast algorithms, associated with the names of Schur and Levinson, are known for the triangular factorization of symmetric, positive definite Toeplitz matrices and their inverses. In this paper we show that these algorithms can be derived from simple arguments involving causality, symmetry, and energy conservation in discrete lossless transmission lines. The results not only provide a nice interpretation of the classical Schur and Levinson algorithms and a certain Toeplitz inversion formula of Gohberg and Semencul, but they also show immediately that the same fast algorithms apply not only to Toeplitz matrices but to all matrices with so-called displacement inertia (1,1). The results have been helpful in suggesting new digital filter structures and in the study of nonstationary second-order processes.