Look-ahead Levinson and Schur algorithms for non-Hermitian Toeplitz systems

Gutknecht, Martin H.; Hochbruck, Marlis

doi:10.1007/s002110050116

Cited by 29 publications

(15 citation statements)

References 27 publications

(80 reference statements)

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“…We do not know any other methods which have been proved to be stable or weakly stable and have worst-case time bound O(n 2 ). Algorithms which involve pivoting and/or look-ahead [18,27,41,43,76] may work well in practice, but seem to require worst-case overhead O(n 3 ) to ensure stability.…”

Section: Resultsmentioning

confidence: 99%

“…From (40), the method is weakly stable (according to Definition 2), although we can not expect the stronger bound (3) to be satisfied. The bounds (40)(41) are similar to those usually given for the method of normal equations [36], not those usually given for the method of semi-normal equations [5,61,66]. This is because, in applications of the semi-normal equations, it is usually assumed thatR is computed via an orthogonal factorization of A, so there is a matrix A such thatR TR = A T A and…”

Section: The Semi-normal Equationsmentioning

confidence: 90%

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Stability analysis of a general toeplitz systems solver

1995

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We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that R T R is close to A T A. Thus, when the algorithm is used to solve the semi-normal equations R T Rx = A T b, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the solution of the full-rank Toeplitz or Hankel least squares problem min Ax − b 2 .

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Section: Resultsmentioning

confidence: 99%

Section: The Semi-normal Equationsmentioning

confidence: 90%

Stability analysis of a general toeplitz systems solver

1995

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“…There are some difficulties, though. The inverse of a Toeplitz matrix does not generally have Toeplitz structure, and the fast factorization algorithms for Toeplitz matrices can require as many as O(n 3 ) flops if pivoting is used to improve stability; see [32,11,4], for example.…”

Section: Introductionmentioning

confidence: 99%

Pivoted Cauchy-Like Preconditioners for Regularized Solution of Ill-Posed Problems

Kilmer¹,

O’Leary²

1999

SIAM J. Sci. Comput.

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Abstract. Many ill-posed problems are solved using a discretization that results in a least squares problem or a linear system involving a Toeplitz matrix. The exact solution to such problems is often hopelessly contaminated by noise, since the discretized problem is quite ill conditioned, and noise components in the approximate null-space dominate the solution vector. Therefore we seek an approximate solution that does not have large components in these directions. We use a preconditioned conjugate gradient algorithm to compute such a regularized solution. A unitary change of coordinates transforms the Toeplitz matrix to a Cauchy-like matrix, and we choose our preconditioner to be a low rank Cauchy-like matrix determined in the course of Gu's fast modified complete pivoting algorithm. We show that if the kernel of the ill-posed problem is smooth, then this preconditioner has desirable properties: the largest singular values of the preconditioned matrix are clustered around one, the smallest singular values, corresponding to the lower subspace, remain small, and the upper and lower spaces are relatively unmixed. The preconditioned algorithm costs only O(n lg n) operations per iteration for a problem with n variables. The effectiveness of the preconditioner for filtering noise is demonstrated on three examples.

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“…These fast algorithms are in general numerically unstable for indefinite systems [3,5,11]. Recently, methods have been proposed [6,10,11] which are numerically stable, but which attempt to retain the O(n 2 ) complexity. However, all of these algorithms will 2 The Gohberg-Kailath-Olshevsky (GKO) Algorithm…”

Section: Introductionmentioning

confidence: 99%

<title>Error analysis of a fast partial pivoting method for structured matrices</title>

Sweet

Brent

1995

SPIE Proceedings

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Many matrices that arise in the solution of signal processing problems have a special displacement structure. For example, adaptive filtering and direction-of-arrival estimation yield matrices of a Toeplitz type. A recent method of Gohberg, Kailath and Olshevsky (GKO) allows fast Gaussian elimination with partial pivoting for such structured matrices. In this paper, a rounding error analysis is performed on the Cauchy and Toeplitz variants of the GKO method. It is shown the error growth depends on the growth in certain auxiliary vectors, the generators, which are computed by the GKO algorithms. It is also shown that in certain circumstances, the growth in the generators can be large, and so the error growth is much larger than would be encountered with normal Gaussian elimination with partial pivoting. A modification of the algorithm to perform a type of row-column pivoting is proposed which may circumvent this problem.

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Look-ahead Levinson and Schur algorithms for non-Hermitian Toeplitz systems

Cited by 29 publications

References 27 publications

Stability analysis of a general toeplitz systems solver

Stability analysis of a general toeplitz systems solver

Pivoted Cauchy-Like Preconditioners for Regularized Solution of Ill-Posed Problems

<title>Error analysis of a fast partial pivoting method for structured matrices</title>

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