A look-ahead algorithm for the solution of general Hankel systems

Freund, Robert M.; Zha, Hongyuan

doi:10.1007/bf01388691

Cited by 41 publications

(13 citation statements)

References 23 publications

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“…This algorithm can be made robust in oating point arithmetic by replacing zero-check conditions with criteria based both on backward error analysis for LU factorization and conditioning estimates for the leading principal submatrices of B. Fast numerical schemes based on similar techniques were developed in [6,15] for the solution of Toeplitz and Hankel linear systems. The generalization of the error analysis presented there to BernsteinBezoutian linear systems is beyond our present scope and it is a part of an ongoing investigation on the numerical properties of resultants for Bernstein polynomials.…”

Section: Introductionmentioning

confidence: 99%

Bernstein–Bezoutian matrices

Бини

Gemignani

2004

Theoretical Computer Science

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Several computational and structural properties of Bezoutian matrices expressed with respect to the Bernstein polynomial basis are shown. The exploitation of such properties allows the design of fast algorithms for the solution of Bernstein-Bezoutian linear systems without never making use of potentially ill-conditioned reductions to the monomial basis. In particular, we devise an algorithm for the computation of the greatest common divisor (GCD) of two polynomials in Bernstein form. A series of numerical tests are reported and discussed, which indicate that Bernstein-Bezoutian matrices are much less sensitive to perturbations of the coe cients of the input polynomials compared to other commonly used resultant matrices generated after having performed the explicit conversion between the Bernstein and the power basis.

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Section: Introductionmentioning

confidence: 99%

Bernstein–Bezoutian matrices

Бини

Gemignani

2004

Theoretical Computer Science

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“…Since row or column pivoting would destroy the Toeplitz structure, some kind of block pivoting must be applied, which one may also call a look-ahead strategy. (For a Hankel system, the look-ahead Lanczos algorithm [18,20] is in fact readily translated into a fast look-ahead Hankel solver [15].) It turns out that there is still much freedom in choosing the details of such block pivoting schemes.…”

Section: Introductionmentioning

confidence: 99%

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

Gutknecht¹,

Hochbruck

1995

Numerische Mathematik

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Summary. We present generalizations of the nonsymmetric Levinson and Schur algorithms for non-Hermitian Toeplitz matrices with some singular or ill-conditioned leading principal submatrices. The underlying recurrences allow us to go from any pair of successive well-conditioned leading principal submatrices to any such pair of larger order. If the look-ahead step size between these pairs is bounded, our generalized Levinson and Schur recurrences require O(N 2 ) operations, and the Schur recurrences can be combined with recursive doubling so that an O(N log 2 N ) algorithm results. The overhead (in operations and storage) of look-ahead steps is very small. There are various options for applying these algorithms to solving linear systems with Toeplitz matrix.Mathematics Subject Classification (1991): 65F05; 30E05, 41A21, 42A70, 60G25

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“…The package will also include implementations of the look-ahead algorithm for the fast solution of general Hankel systems described in [15] as well as of a Schur-type look-ahead Hankel solver.…”

Section: Discussionmentioning

confidence: 99%

A look-ahead Bareiss algorithm for general Toeplitz matrices

Freund

1994

Numerische Mathematik

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The Bareiss algorithm is one of the classical fast solvers for systems of linear equations with Toeplitz coefficient matrices. The method takes advantage of the special structure, and it computes the solution of a Toeplitz system of order N with only O(N 2 ) arithmetic operations, instead of O(N 3 ) operations. However, the original Bareiss algorithm requires that all leading principal submatrices be nonsingular, and the algorithm is numerically unstable if singular or ill-conditioned submatrices occur. In this paper, an extension of the Bareiss algorithm to general Toeplitz systems is presented. Using look-ahead techniques, the proposed algorithm can skip over arbitrary blocks of singular or ill-conditioned submatrices, and at the same time, it still fully exploits the Toeplitz structure. Implementation details and operations counts are given, and numerical experiments are reported. We also discuss special versions of the proposed look-ahead Bareiss algorithm for Hermitian indefinite Toeplitz systems and banded Toeplitz systems. Subject Classification (1991): 65F05, 15A57, 42C05 Mathematics

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A look-ahead algorithm for the solution of general Hankel systems

Cited by 41 publications

References 23 publications

Bernstein–Bezoutian matrices

Bernstein–Bezoutian matrices

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

A look-ahead Bareiss algorithm for general Toeplitz matrices

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