TY - JOUR

T1 - Inversion and factorization of non-Hermitian quasi-Toeplitz matrices

AU - Bistritz, Yuval

AU - Kailath, Thomas

N1 - Funding Information:
*Research supported in part by the Air Force Office of Scientific Research, Air Force Systems Command, under Contract AF83-0228; by the Department of the Navy, Office of Naval Research, under contract N9091485K9612; by the US. Army Research Office, under Contract DAALO3-8SK-0045; and by the Air Force Office of Scientific Research, Air Force. Yuval Bistritz also gratefully acknowledges the support by a Chaim Weizmann postdoctoral fellowship award. ‘NOW with AT&T Bell Laboratories, Murray Hill, New Jersey 07974.

PY - 1988/1

Y1 - 1988/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=45549114290&partnerID=8YFLogxK

U2 - 10.1016/0024-3795(88)90161-9

DO - 10.1016/0024-3795(88)90161-9

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AN - SCOPUS:45549114290

SN - 0024-3795

VL - 98

SP - 77

EP - 121

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - C

ER -