Inversion and factorization of non-Hermitian quasi-Toeplitz matrices

Abstract

This paper considers formulas and fast algorithms for the inversion and factorization of non-Hermitian Toeplitz and quasi-Toeplitz (QT) matrices (matrices with a certain “hidden” Toeplitz structure). The results include the following generalizations: (1) A Schur algorithm that extends to non-Hermitian matrices a previous triangular factorization algorithm for Hermitian QT matrices. (2) A Levinson algorithm that generalizes to non-Hermitian matrices a previous Levinson algorithm that finds the triangularly factorized inverses of certain (so-called admissible) QT matrices. (3) The extension to QT matrices of the Gohberg-Semencul (GS) inversion formula for non-Hermitian Toeplitz matrices. Next, the paper introduces a new fast algorithm, called the extended QT factorization algorithm, that overcomes the restriction to admissibility matrices of the above Levinson algorithm. The new algorithm is efficient and comprehensive; it produces, for a general QT matrix R n of size ( n + 1)×( n + 1), the triangularly factorized inverses and the GS type inverses of the matrix and all its submatrices, as well as the triangular factorization of R n itself, all in approximately 7 n 2 elementary operations for a non-Hermitian and 3.5 n 2 for a Hermitian matrix. The fast algorithms for non-Hermitian QT matrices are shown to be associated with two discrete transmission lines (which reduce to the familiar single lattice in the Hermitian case). All the presented algorithms are illustrated and interpreted in terms of input sequences and flows of signals in the related transmission line realization.