I. Gohberg, T. Kailath, and V. Olshevsky. Gaussian elimination with partial pivoting for structured matrices. Technical report, Information Systems Lab., Stanford University, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....algorithms to solve structured matrices such as Toeplitz and Hankel matrices have been proposed and studied over the last couple of decades. A fundamental problem with these fast methods is that they do not allow any form of pivoting because the structure of the matrices is destroyed. In [12] [11] the authors suggest ways to overcome this problem by transforming one class of structured matrices to another using fast trigonometric transforms in such a way that pivoting may be incorporated in the factorization algorithms. These algorithms factor Toeplitz, Hankel and Vandermonde matrices by ....

....Hankel and Vandermonde matrices by converting them to Cauchy matrices and performing Gaussian elimination with partial pivoting. It was shown that the special displacement structure of Cauchy matrices is conducive to pivoting strategies such as partial pivoting. The algorithms suggested in [12] [11], however, did not exploit properties such as realness and symmetry simultaneously in the matrices. Recently there has been a surge of activity in this area and new variants are constantly being developed to solve various structured matrix problems with pivoting. This paper attempts to collect all ....

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I. Gohberg, T. Kailath, and V. Olshevsky. Gaussian elimination with partial pivoting for structured matrices. Technical report, Information Systems Lab., Stanford University, 1994.

....only when the look ahead step size is small. For large look ahead steps of size m, the complexity to invert the look ahead block (which is O(m 3 ) makes look ahead algorithms quite expensive. A novel idea to overcome this problem was suggested by Heinig [37] and Gohberg, Kailath and Olshevsky [38]. They converted indefinite Toeplitz and Hankel matrices to Cauchy like matrices using fast trigonometric transforms such as the discrete Fourier transform (DFT) The displacement structure of Cauchy like matrices is invariant to permutation. This allows pivoting to be incorporated into the ....

....The displacement structure of Cauchy like matrices is invariant to permutation. This allows pivoting to be incorporated into the factorization algorithms. This idea was used to develop pivoted factorization algorithms for indefinite Toeplitz and Hankel matrices. One drawback of the algorithms in [37, 38] was that Hermitian Toeplitz matrices were converted to non Hermitian Cauchy like matrices prior to factorization. We overcome this problem with an algorithm to update the generators of a Hermitian form of the displacement equation for Hermitian Cauchy like matrices [24] Another algorithm that ....

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I. Gohberg, T. Kailath, and V. Olshevsky, "Gaussian elimination with partial pivoting for structured matrices," tech. rep., Information Systems Lab., Stanford University, 1994.

....a low rank approximation of the Toeplitz matrix. Finally we discuss algorithms to factor Toeplitz matrices by converting them to Cauchy type matrices. Toeplitz matrices can be converted using the discrete Fourier transform into Cauchy type matrices that allow pivoting during the factorization [21, 15]. These algorithms also have the same complexity,O(n 2 ) as the Schur algorithm. The problem with this method is that any real valued Toeplitz matrix is converted to a complex Cauchy type matrix and the entire factorization algorithm proceeds in complex arithmetic. This is computationally ....

....to this algorithm that allows us to work in real arithmetic and also exploit the symmetric structure of the matrix. This yields a rank revealing algorithm for the factorization of a semidefinite block Toeplitz matrix that is computationally less expensive than the algorithm presented in [21, 15]. 2 Symmetric positive definite block Toeplitz matrices In this section we present a block generalization of the classical Schur algorithm [8, 9] using block hyperbolic Householder reflectors. Block hyperbolic Householder transformations can be applied at the BLAS 3 rate rather than plain ....

[Article contains additional citation context not shown here]

I. Gohberg, T. Kailath, and V. Olshevsky, Gaussian elimination with partial pivoting for structured matrices, tech. rep., Information Systems Lab., Stanford University, 1994.

....the introduction of pivoting or look ahead (with block steps) in the Bareiss and Levinson algorithms [18, 19, 28, 29, 30, 75, 76] and this is often successful in practice, but in the worst case the overhead is O(n 3 ) operations. The recent algorithm GKO of Gohberg, Kailath and Olshevsky [34] may be as stable as Gaussian elimination with partial pivoting, but an error analysis has not been published. In an attempt to achieve stability without pivoting or look ahead, it is natural to consider algorithms for computing an orthogonal factorization A = QR (1) of A. The first such O(n 2 ....

....algorithm given in x3 of [8] which computes Q explicitly. Thus, considering both speed and stability, it is best to avoid the computation of Q. For the method of Nagy [60] the multiplication count is 16n 2 O(n) and for Cybenko s method [23] it is 23n 2 O(n) The method TpH of [34] requires 21n 2 =2 O(n log n) real multiplications, and the method GKO of [34] requires 13n 2 =2 O(n log n) complex multiplications. Thus, the method of x6 should be faster than any of these methods, although GKO may be competitive if the Toeplitz matrix A complex. For the rectangular case ....

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I. Gohberg, T. Kailath and V. Olshevsky, Gaussian Elimination with Partial Pivoting for Structured Matrices, preprint, 14 May 1994.

....a low rank approximation of the Toeplitz matrix. Finally we discuss algorithms to factor Toeplitz matrices by converting them to Cauchy type matrices. Toeplitz matrices can be converted using the discrete Fourier transform into Cauchy type matrices that allow pivoting during the factorization [21, 15]. These algorithms also have the same complexity,O(n 2 ) as the Schur algorithm. The problem with this method is that any real valued Toeplitz matrix is converted to a complex Cauchy type matrix and the entire factorization algorithm proceeds in complex arithmetic. This is computationally ....

....to this algorithm that allows us to work in real arithmetic and also exploit the symmetric structure of the matrix. This yields a rank revealing algorithm for the factorization of a semidefinite block Toeplitz matrix that is computationally less expensive than the algorithm presented in [21, 15]. 3 2. Symmetric positive definite block Toeplitz matrices In this section we present a block generalization of the classical Schur algorithm [8, 9] using block hyperbolic Householder reflectors. Block hyperbolic Householder transformations can be applied at the BLAS 3 rate rather than plain ....

[Article contains additional citation context not shown here]

I. Gohberg, T. Kailath, and V. Olshevsky. Gaussian elimination with partial pivoting for structured matrices. Technical report, Information Systems Lab., Stanford University, 1994.

....problems. The rank revealing QR factorization algorithm is based on adapting the generalized Schur algorithm to Cauchy like matrices while the second algorithm is based on adapting the augmented systems method to Toeplitz matrices. Both algorithms are based on the theory of Gohberg et al. [6] and Heinig [10] that structured matrices such as Toeplitz, Hankel and Vandermonde may be converted to Cauchy like matrices such that fast pivoted factorization may be made possible without destroying the displacement structure. In [5] Gallivan et al. present a modification of the generalized ....

I. Gohberg, T. Kailath, and V. Olshevsky. Gaussian elimination with partial pivoting for structured matrices. Technical report, Information Systems Lab., Stanford University, 1994.

....operations in the worst case and can be shown to be weakly stable, but not stable in the usual sense of backward error analysis. Thus there is an interest in fast algorithms which require O(n 2 ) operations in the worst case and can be shown to be stable. Recently, Gohberg, Kailath and Olshevsky [8] have shown how to perform Gaussian elimination in a fast way with matrices with a special displacement structure. Such matrices include Toeplitz, Vandermonde, Hankel and Cauchy matrices, and generalizations thereof, called Toeplitz type, etc. They also show how to incorporate partial pivoting ....

....we first define the displacement operator, displacement equation and displacement rank for structured matrices; we then give the general Gaussian elimination algorithm for structured matrices, followed by the variants for Cauchy and Toeplitz matrices. 2. 1 Displacement structure Gohberg et al. [8] show that structured matrices satisfy a Sylvester equation which has the form r fA f ;A b g (R) A f R Gamma RA b = Phi Psi ; 1) where A f and A b have some simple structure (usually banded, with 3 or fewer full diagonals) Phi and Psi are n Theta ff and ff Theta n respectively, and ff ....

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I. Gohberg, T. Kailath and V. Olshevsky, "Gaussian elimination with partial pivoting for structured matrices", preprint, May 1994.

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I. Gohberg, T. Kailath and V. Olshevsky, "Gaussian Elimination with Partial Pivoting for Structured Matrices ", preprint

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I. Gohberg, T. Kailath and V. Olshevsky, "Gaussian elimination with partial pivoting for structured matrices", Math. Comp. 64 (1995), 1557--1576.

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