TY - JOUR

T1 - Immittance- versus scattering-domain fast algorithms for non-Hermitian Toeplitz and quasi-Toeplitz matrices

AU - Bistritz, Yuval

AU - Lev-Ari, Hanoch

AU - Kailath, Thomas

N1 - Funding Information:
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 twoterm 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 immittunce-dotnuin variables. This paper provides extensions for previous immittance Schur and Levinson algorithms from Hermitian to non-Hermitian matrices. It consid- *Research supported in part by the Air Force Office of Scientific Research, Air Force Systems Command, under Contract AF-830228, by the Department of the Navy, Office of Naval Research, under Contract NOOO1485K~612, by the U.S. Army Research Office, under Contract DAAL03-88K@O45, and by the Air Force Office of Scientific Research, Air Force. Yuvai Bistritz also gratefully acknowledgest he support of a Chaim Weizmann postdoctoral fellowship award.

PY - 1989

Y1 - 1989

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=45249127923&partnerID=8YFLogxK

U2 - 10.1016/0024-3795(89)90678-2

DO - 10.1016/0024-3795(89)90678-2

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AN - SCOPUS:45249127923

SN - 0024-3795

VL - 122-124

SP - 847

EP - 888

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - C

ER -