T. Kailath and A. H. Sayed. Fast Reliable Algorithms for Matrices with Structure. SIAM, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....matrices is the Schur algorithm, which has been shown to be backward stable in the case that the matrix is positive definite. We describe briefly this special case since it is the cornerstone for all variants described in this report. A more elaborate description of the algorithm can be found in [4]. We start with the matrix rT = T Gamma Z TZ; Z 6 6 6 6 0 1 : 0 0 . 1 0 : 0 0 7 7 7 7 The matrix rT is called the displacement of T and clearly has low rank since rT = 6 6 6 t 1 0 : 0 . t n Gamma1 0 : 0 7 7 7 : 1) ....

.... equations (3) 4) imply that the first row of U is given by the first row of G : U(1; G(1; A recursive algorithm for the successive rows of U is then obtained if one can construct the generator G for the Schur complement of T : T Gamma U(1; U(1; G: 5) It is shown in [4] that a simple shift operation with Z applied to the first row of G does the job : G 0 x 1 : x n Gamma1 0 y 1 : y n Gamma1 0 x 0 : x n Gamma2 G(1; Z G(2; 6) Eliminating the element y 1 in G then yields the second row of U and so on. This is ....

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T. Kailath, A. Sayed, Fast Reliable Algorithms for Matrices with Structure SIAM Publ., Philadelphia, 1999.

....matrices is the Schur algorithm, which has been shown to be backward stable in the case that the matrix is positive definite. We describe briefly this special case since it is the cornerstone for all variants described in this report. A more elaborate description of the algorithm can be found in [3]. We start with the matrix rT : T Gamma Z T TZ; Z : 2 6 6 6 6 4 0 1 : 0 0 . 1 0 : 0 0 3 7 7 7 7 5 : The matrix rT is called the displacement of T and clearly has low rank since rT = 2 6 6 6 4 t 0 t 1 : t n Gamma1 t 1 0 : 0 . ....

.... imply that the first row of U is given by the first row of G : U(1; G(1; A recursive algorithm for the successive rows of U is then obtained if one can construct the generator G for the Schur complement of T : T Gamma U(1; T U(1; G T Sigma G: 5) It is shown in [3] that a simple shift operation with Z applied to the first row of G does the job : G : 0 x 1 : x n Gamma1 0 y 1 : y n Gamma1 : 0 x 0 : x n Gamma2 0 y 1 : y n Gamma1 = G(1; Z G(2; 6) Eliminating the element y 1 in G then yields the second ....

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T. Kailath, A. Sayed, Fast Reliable Algorithms for Matrices with Structure SIAM Publ., Philadelphia, 1999.

....null space of a matrix, finite fields. 1991 Math. Subject Classification: 47B35, 68Q40, 65F05, 15A23, 15A33, 68Q25. 1 Introduction 1. 1 Computations with structured matrices The study of structured matrices is a classical subject with long history (cf. C841] Ho70] KVM78] KS95] [KS99]) Such matrices are omnipresent in signal and image processing, sciences and engineering. Some of the best known classical structured matrices are displayed in Table 1. Supported by NSF Grant CCR 9732206 and PSC CUNY Award 669363 Table 1 Toeplitz matrices: Hankel matrices: T = t i Gammaj ) ....

.... also with displacement operators) and are very popular and well recognized too, in particular for their application to celestial mechanics and algebraic decoding and their correlation to polynomial and rational interpolation and multipoint evaluation (see [PLST93] PZHY97] PACLS98] OP98] [KS99], OS99] OP99] and bibliography therein) There are various important classes of structured matrices, such as Bezoutians, Loewner, Pick, Sylvester and subresultant matrices, closely related to the above four classes. They are also associated with linear operators of displacement and or scaling ....

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T. Kailath, A. Sayed (editors), Fast Reliable Algorithms for Matrices with Structure, SIAM Publications, Philadelphia, 1999.

....representation of a holomorphic function in the unit disk [34] This class can be generalized to cover unsymmetric matrices and more general low displacement rank matrices [28] In this paper we consider the numerical stability of some of these algorithms. A more detailed survey is given in [29]. In the following, R denotes a structured matrix, T is a Toeplitz or Toeplitztype matrix, P is a permutation matrix, L is lower triangular, U is upper triangular, and Q is orthogonal. In error bounds, On ( means O( f(n) where f(n) is a polynomial in n. 1 Asymptotically faster algorithms ....

....the generalized Schur algorithm. This was considered by M. Stewart and Van Dooren [36] and by Chandrasekharan and Sayed [12] The results were obtained independently by the two pairs of authors, and the generalized Schur algorithm considered in each case is slightly different for details see [29]. The conclusion is that the generalized Schur algorithm is stable for positive definite symmetric (or Hermitian) matrices, provided that the hyperbolic transformations in the algorithm are implemented correctly. In contrast, BBHS showed that stability of the classical Schur Bareiss algorithm is ....

T. Kailath and A. H. Sayed (editors), Fast Reliable Algorithms for Matrices with Structure, to appear.

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T. Kailath and A. H. Sayed, editors. Fast Reliable Algorithms for Matrices with Structure, SIAM, PA, 1999

....words: scattering, displacement structure, Schur algorithm, matrix extension, matrix inequality, tensor algebra. 1 The Generalized Schur Algorithm The displacement structure concept provides a convenient framework for the study of matrix problems involving different kinds of structure (see, e.g. [1, 2, 3]) In this paper we use the displacement structure formalism, and the related Schur algorithm, to elaborate on a generalized version of certain classical matrix extension problems. We highlight connections with structured matrix inequalities and also relate the discussion to tensor algebra. Both ....

T. Kailath and A. H. Sayed, editors, Fast Reliable Algorithms for Matrices with Structure, SIAM, PA, 1999.

....Merched was also supported by a fellowship from CAPES, Brazil. R. Merched and A. H. Sayed are with the Department of Electrical Engineering, University of California, Los Angeles, CA 90095. the new extensions rather immediately, by exploiting different kinds of matrix structure (e.g. 5] 6] [7]) The MDF schemes of this paper are attractive for applications where real arithmetic is required. Moreover, since efficient algorithms exist for computing the DCT, DST, and DHT (see, e.g. 8] these schemes also lead to efficient adaptive filter structures. We shall further present in Sec. IX ....

.... is known that C III diagonalizes any K Theta K structured matrix functions A(z) that can be expressed as the sum of Toeplitzplus Hankel matrix functions in the following form (this fact is developed in [5] in the context of constant matrices with socalled displacement structure (see, e.g. 6] [7]) A(z) T(z) H(z) B(z) 23) where T(z) is a symmetric Toeplitz matrix, H(z) is a Hankel matrix related to T(z) and B(z) is a border matrix also related to T(z) For example, for K = 4, fA(z) B(z) H(z)g have the 2 The derivation applies equally well to other trigonometric transforms, ....

T. Kailath and A. H. Sayed, editors, Fast Reliable Algorithms for Matrices with Structure, SIAM, PA, 1999.

....90095. x Electrical Engineering Dept. University of California, Los Angeles, CA 90095. of T 0 satisfies a similar displacement equation, viz. T Gamma1 0 Gamma F T T Gamma1 0 F = H 0 JH T 0 ; with the same signature matrix J but with the roles of fF; F T g reversed (see, e.g. [1, 2]) Now assume that for successive time instants k 0 the values of fT k ; b k g are obtained recursively as follows: T k 1 = T k a k a T k ; b k 1 = b k j(k)a k ; for some known column vectors fa k g and scalars fj(k)g. In other words, T k 1 is a rank one update of T k and b k 1 is ....

....Equations (FUS) 1. Initialization (overhead costs) Given fG 0 ; H 0 g, compute x 0 and r 0 using G 0 and the so called generalized Schur algorithm with back substitution, or by using H 0 and fast matrix vector multiplication procedures for matrices with displacement structure (see, e.g. [1, 2, 4, 5]) Compute also fi(0) as an inner product. 2. Assume that we have fH k ; r k ; x k ; fi(k)g and iterate: ffl x k 1 = x k fi(k)r k [j(k) Gamma a T k x k ] O(n) flops] ffl Compute the QR factorization (2) O(nff 2 l 1 ) flops] ffl Compute the eigendecomposition (3) O(ff 3 ) ....

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T. Kailath and A. H. Sayed, eds., Fast Reliable Algorithms for Matrices with Structure, SIAM, PA, 1999.

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T. Kailath and A. H. Sayed. Fast Reliable Algorithms for Matrices with Structure. SIAM, 1999.

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T. Kailath, A. H. Sayed (eds. ): Fast Reliable Algorithms for Matrices with Structure. SIAM Press, Philadelphia, 1999.

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T. Kailath and A. Sayed, Fast reliable algorithms for matrices with structure. SIAM, Philadelphia, 1999.

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Kailath T. and Sayed A. H., editors, Fast reliable algorithms for matrices with structure. SIAM, Philadelphia, 1999.

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T. Kailath and A. H. Sayed, editors, "Fast reliable algorithms for matrices with structure," SIAM, Philadelphia, 1999.

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T. Kailath and A. Sayed, Fast reliable algorithms for matrices with structure. SIAM, Philadelphia, 1999.

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