Immittance- versus scattering-domain fast algorithms for non-Hermitian Toeplitz and quasi-Toeplitz matrices

Yuval Bistritz*, Hanoch Lev-Ari, Thomas Kailath

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

The classical algorithms of Schur and Levinson are efficient procedures to solve sets of Hermitian Toeplitz linear equations or to invert the corresponding coefficient matrices. They propagate pairs of variables that may describe incident and scattered waves in an associated cascade-of-layered-media model, and thus they can be viewed as scattering-domain algorithms. It was recently found that a certain transformation of these variables followed by a change from two-term to three-term recursions results in reduction in computational complexity in the abovementioned algorithms roughly by a factor of two. The ratio of such pairs of transformed variables can be interpreted in the above layered-media model as an impedance or admittance; hence the name immittance-domain variables. This paper provides extensions for previous immittance Schur and Levinson algorithms from Hermitian to non-Hermitian matrices. It considers both Toeplitz and quasi-Toeplitz matrices (matrices with certain "hidden" Toeplitz structure) and compares two- and three-term recursion algorithms in the two domains. The comparison reveals that for non-Hermitian matrices the algorithms are equally efficient in both domains. This observation adds new comprehension to the source and value of algorithms in the immittance domain. The immittance algorithms, like the scattering algorithms, exploit the (quasi-)Toeplitz structure to produce fast algorithms. However, unlike the scattering algorithms, they can respond also to symmetry of the underlying matrix when such extra structure is present, and yield algorithms with improved efficiency.

Original languageEnglish
Pages (from-to)847-888
Number of pages42
JournalLinear Algebra and Its Applications
Volume122-124
Issue numberC
DOIs
StatePublished - 1989

Funding

FundersFunder number
Department of the Navy
U.S. Army Research Office
Office of Naval ResearchNOOO1485K~612
Air Force Office of Scientific Research
Air Force Materiel CommandAF-830228
Air Force Institute of Technology

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