J. Chun and T. Kailath, Divide--and--conquer solutions of least squares problems for matrices with displacement structure, SIAM J. Matrix Anal. Appl., 12 (1991), pp. 128--142.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is need for computing solutions in near real time , considerable effort has been devoted to developing fast algorithms for the solution of (1. 1) Most of this work has focused on direct methods, such as the fast QR factorization algorithms of Bojanczyk, Brent and de Hoog [4] Chun and Kailath [11], Cybenko [13] and Sweet [26] The stability properties of these algorithms are not well understood, see Bunch [5] for nonsingular Toeplitz systems, and Luk and Qiao for direct orthogonal factorization for least squares problems [22] Almost all fast orthogonal factorization methods involve the ....

....structure representation of a matrix. We introduce the n Theta n lower shift matrix Z, whose entries are zero everywhere except for 1 s on the first subdiagonal. The displacement operator r is defined by rA = A Gamma ZAZ where rA is called the displacement of A, cf. Chun and Kailath [11]. Let L(w) denote the n Theta n lower triangular Toeplitz matrix with first column the vector w. Using these definitions, the following lemma can be proved [12] Lemma 2.1. An arbitrary n Theta n matrix A can be written in the form ae i=1 L(u i )L (v i ) where ae = rank(rA) and u i ....

J. Chun and T. Kailath, Divide--and--conquer solutions of least squares problems for matrices with displacement structure, SIAM J. Matrix Anal. Appl., 12 (1991), pp. 128--142.

....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 ....

J.Chun and T.Kailath, Divide-and-conquer solution of least-squares problems for matrices with displacement structure, SIAM J. Matrix Anal. Appl. (submitted).

....multiplication for three classes of matrices whose properties relate to the properties of Toeplitz, Vandermonde and Cauchy matrices and also for their inverses. The definitions of these classes of matrices are based on the concept of displacement. Let F f and F b be matrices from C . Following [6] let us refer to the matrix r fF f ;F b g (A) A Gamma F f AF b (0.1) as fF f ; F b g displacement of the matrix A 2 C . The number ff = rankr fF f ;F b g (A) is called fF f ; F b g displacement rank of the matrix A 2 C . Each pair of rectangular matrices G and H such that r fF f ;F b g ....

....low fZ 0 ; Z 0 g displacement rank are called close to Toeplitz (they are also referred to as near to Toeplitz, Toeplitz like, of the Toeplitz type, etc. The classes of close to Vandermonde and close to Cauchy matrices (the names Vandermonde like, Cauchylike are also used) are defined (see [13] [6]) using another displacement operator 4 fF f ;F b g (A) F f A Gamma AF b : Namely, matrix A is called close to Vandermonde if it is transformed by 4 fdiag(t) Z 1 g ( Delta) in the matrix of low rank. Furthermore, in the case when rank of the matrix 4 fdiag(q) diag(t)g (A) is ....

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Chun J. and Kailath T., Divide-and-conquer solutions of least-squares problems for matrices with displacement structure, SIAM Journal of Matrix Analysis Appl., 12 (No. 1) : 128 - 145 (1991).

....is need for computing solutions in near real time , considerable e#ort has been devoted to developing fast algorithms for the solution of (1. 1) Most of this work has focused on direct methods, such as the fast QR factorization algorithms of Bojanczyk, Brent and de Hoog [4] Chun and Kailath [11], Cybenko [13] and Sweet [26] The stability properties of these algorithms are not well understood, see Bunch [5] for nonsingular Toeplitz systems, and Luk and Qiao for direct orthogonal factorization for least squares problems [22] Almost all fast orthogonal factorization methods involve the ....

....structure representation of a matrix. We introduce the n n lower shift matrix Z, whose entries are zero everywhere except for 1 s on the first subdiagonal. The displacement operator # is defined by #A = A ZAZ # , where #A is called the displacement of A, cf. Chun and Kailath [11]. Let L(w) denote the nn lower triangular Toeplitz matrix with first column the vector w. Using these definitions, the following lemma can be proved [12] Lemma 2.1. An arbitrary n n matrix A can be written in the form A = # X i=1 L(u i )L # (v i ) where # = rank(#A) and u i and v i are ....

J. Chun and T. Kailath, Divide--and--conquer solutions of least squares problems for matrices with displacement structure, SIAM J. Matrix Anal. Appl., 12 (1991), pp. 128--142.

.... 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 ....

J.Chun and T.Kailath, Divide-and-conquer solution of least-squares problems for matrices with displacement structure, SIAM J. Matrix Anal. Appl. (submitted).

....is a Vandermonde matrix, one may assert that V V T is a Hankel matrix. Hence the complexity for solving the considered system is the same as for solving a Hankel linear system. Since there are O(nlog 2 n) algorithms for computing the inverse of a positive definite Hankel matrix (see [3] 6] [8], 19] among others) one finds out another way of constructing O(nlog 2 n) algorithms for the polynomial interpolation. On the contrary, when attempting to extend this technique to block Vandermonde matrices, the difficulty arises as soon as one writes the structure: V T Z p Gamma D T ....

.... exploit the O(np(lognp)logn) fast method of [17] for computing the products V AV T and V W b, and on the other hand it is possible to solve the symmetric positive definite and Toeplitz like system V AV T x = V W b with O(mlog 2 m) operations only appealing the rapid techniques developed in [8]. Hence, O(np(lognp)logn mlog 2 m) arithmetic operations are sufficient for solving the just introduced least squares minimization problem. 7. Confluent Vandermonde least squares minimizations. In this section, we present another application of the Sylvester s structure of the m Theta np; m ....

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J. Chun and T. Kailath. Divide and conquer solutions of least squares problems for matrices with displacement structure SIAM J. Matrix. Anal. Appli. 12(1991) pp.128-145

....were announced in Parallel algorithms and numerical stability for Toeplitz systems, invited paper presented by Brent at the SIAM Conference on Linear Algebra in Signals, Systems and Control, Seattle, August 16 19, 1993. rpb143tr typeset using L a T E X Asymptotically faster algorithms exist [1, 4, 14, 20, 40, 47, 58, 59]. Sometimes these algorithms are called superfast [1] We avoid this terminology because, even though the algorithms require only O(n(log n) 2 ) arithmetic operations, they may be slower than O(n 2 ) algorithms for n 256 (see [1, 20, 68] We prefer the term asymptotically fast. The ....

.... faster algorithms exist [1, 4, 14, 20, 40, 47, 58, 59] Sometimes these algorithms are called superfast [1] We avoid this terminology because, even though the algorithms require only O(n(log n) 2 ) arithmetic operations, they may be slower than O(n 2 ) algorithms for n 256 (see [1, 20, 68]) We prefer the term asymptotically fast. The numerical stability properties of asymptotically fast algorithms are generally either bad [16] or unknown, although some positive partial results have been obtained recently [42] Attempts to stabilise asymptotically fast algorithms by look ahead ....

J. Chun and T. Kailath, "Divide-and-conquer solutions of least-squares problems for matrices with displacement structure", SIAM J. Matrix Anal. Appl. 12 (1991), 128-145.

....0 0 0 . 1 0 . 0 . 0 0 0 Delta Delta Delta 1 0 3 7 7 7 7 7 7 7 5 : The displacement operator r is defined by rA n = A n Gamma Z n A n Z n ; 16 Fast Reliable Algorithms for Matrices with Structure where rA n is called the displacement of A n , cf. Chun and Kailath [40]. Let L n (w) denote the n Theta n lower triangular Toeplitz matrix with first column the vector w. Theorem 3.3.1. J. Chun, T. Kailath, and H. Lev Ari [41] An arbitrary n Theta n matrix A n can be written in the form A n = ae X i=1 L n (u i )L n (v i ) 3.25) where ae = rank(rA n ) ....

J. Chun and T. Kailath, Divide-and-Conquer Solutions of Least-Squares Problems for Matrices with Displacement Structure, SIAM J. Matrix Anal. Appl., 12 (1991), pp. 128--145.

....areas, such as the denoising problems, see T. Chan and Olkin [59] Considerable effort has been devoted to developing fast algorithms for them and most works have been focused on direct methods, such as the fast QR factorization algorithms of Bojanczyk, Brent, and de Hoog [18] Chun and Kailath [63], Cybenko [68] and Sweet [176] Here we will consider using the preconditioned conjugate gradient method. Although the classical conjugate gradient algorithm applies only to square Hermitian positive definite systems, one can still use it to find the solution to (4.13) by applying it to the ....

....structure representation of Toeplitz matrices. We introduce the n by n lower shift matrix Z, whose entries are zero everywhere except for 1 s on the first subdiagonal. The displacement operator r is defined by rA = A Gamma ZAZ ; where rA is called the displacement of A, see Chun and Kailath [63]. Let L(w) denote the n by n lower triangular Toeplitz matrix with the vector w as its first column. Using these definitions, we have the following lemma. Theorem 24 (Chun, Kailath, and Lev Ari (1987) 64] An arbitrary n by n matrix A can be written in the form A = ae X i=1 L(u i )L(v i ....

J. Chun and T. Kailath, Divide-and-Conquer Solutions of Least-Squares Problems for Matrices with Displacement Structure, SIAM J. Matrix Anal. Appl., 12 (1991), pp. 128-- 145.

....areas, such as the denoising problems, see T. Chan and Olkin [58] Considerable effort has been devoted to developing fast algorithms for them and most works have been focused on direct methods, such as the fast QR factorization algorithms of Bojanczyk, Brent, and de Hoog [17] Chun and Kailath [62], Cybenko [67] and Sweet [176] Here we will consider using the preconditioned conjugate gradient method. Although the classical conjugate gradient algorithm applies only to square Hermitian positive definite systems, one can still use it to find the solution to (4.13) by applying it to the ....

....structure representation of Toeplitz matrices. We introduce the n by n lower shift matrix Z, whose entries are zero everywhere except for 1 s on the first subdiagonal. The displacement operator r is defined by rA = A Gamma ZAZ ; where rA is called the displacement of A, see Chun and Kailath [62]. Let L(w) denote the n by n lower triangular Toeplitz matrix with the vector w as its first column. Using these definitions, we have the following lemma. Theorem 24 (Chun, Kailath, and Lev Ari (1987) 63] An arbitrary n by n matrix A can be written in the form A = ae X i=1 L(u i )L(v i ....

J. Chun and T. Kailath, Divide-and-Conquer Solutions of Least-Squares Problems for Matrices with Displacement Structure, SIAM J. Matrix Anal. Appl., 12 (1991), pp. 128-- 145.

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Chun,J. and Kailath,T. (1991), Divide-and-conquer solutions of least-squares problems for matrices with displacement structure, SIAM Journal of Matrix Analysis Appl., 12 (No. 1), 128 -- 145. 14

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