Inversion and factorization of non-Hermitian quasi-Toeplitz matrices

Yuval Bistritz*, Thomas Kailath

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

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 Rn 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 Rn itself, all in approximately 7n2 elementary operations for a non-Hermitian and 3.5n2 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.

Original languageEnglish
Pages (from-to)77-121
Number of pages45
JournalLinear Algebra and Its Applications
Volume98
Issue numberC
DOIs
StatePublished - Jan 1988
Externally publishedYes

Funding

FundersFunder number
Department of the Navy
Office of Naval ResearchN9091485K9612, DAALO3-8SK-0045
Air Force Office of Scientific Research
Air Force Materiel CommandAF83-0228
Air Force Institute of Technology

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