T.Kailath and J.Chun, Generalized Gohberg-Semencul formulas for matrix inversion, Op. Theory: Advan. Appl. 40 (1989), 231--246.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with bounded displacement rank. These matrices can be stored compactly by representing them by their displacement generators. As an additional benefit, the use of displacement generators gave fast sequential algorithms for the inversion and factorization of such matrices (see [14] 59] 2] 23] [44]) These so called Schur algorithms for matrices with constant displacement rank run e#ciently on sequential machines. Their implementation on parallel machines has been extensively studied by Kailath and his many coworkers and PhD students. Unfortunately, it is not clear how to parallelize the ....

T.Kailath and J.Chun, Generalized Gohberg-Semencul formulas for matrix inversion, Op. Theory: Advan. Appl. 40 (1989), 231--246.

....0024 3795 94 6.00 2 [17] who outlined a general technique that is in particular applicable to other matrices with small displacement rank, but also to Hankel matrices. See also Friedlander, Morf, Kailath, and Ljung [10] and, ten years later, Heinig and Rost [18, 19] and Kailath and Chun [20] for similar generalizations. However, this is just a very limited selection of the work on inversion formulas for Toeplitz and related matrices. In particular, there are also a number of papers on generalizations to block matrices, which are not addressed here. In an equivalent recursive form, ....

T. Kailath and J. Chun, Generalized Gohberg-Semencul formulas for matrix inversion, vol. 40 of Operator Theory: Advances and Applications, Birkhauser, Basel, 1989, pp. 231--246.

.... we assume that for every input matrix with displacement rank at most ffi, a displacement generator is also available, so an n Theta n matrix with displacement rank at most ffi will be stored in space 2nffi: As an additional benefit, the use of displacement generators gave fast algorithms (see [4, 6, 31, 31, 3, 8, 20]) for basic operations on structured matrices. Lemma 1.1. see [4, 6, 32] a) The product of an n Theta n Toeplitz matrix with an n vector is computable in parallel time O(log n) with P (n) processors. b) The product of two n Theta n Toeplitz matrices is computable (i.e. represented as a ....

T.Kailath and J.Chun, Generalized Gohberg-Semencul formulas for matrix inversion, Op. Theory: Advan. Appl. 40 (1989), 231--246.

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