A Weakly Stable Algorithm for Padé Approximants and the Inversion of Hankel Matrices

Cabay, Stan; Meleshko, Ron

doi:10.1137/0614053

Cited by 49 publications

(36 citation statements)

References 46 publications

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“…However, none of them is exactly a generalization of either the Schur or the Levinson algorithm in the sense that it reduces to one of these classical algorithms in the absence of look-ahead steps. The underlying idea for the methods in [19] is analogous to the one of Cabay and Meleshko's weakly stable diagonal Padé recurrences and the corresponding look-ahead Hankel solver [8,28].…”

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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Section: Introductionmentioning

confidence: 99%

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

Gutknecht¹,

Hochbruck

1995

Numerische Mathematik

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“…Recently, Cabay and Meleshko [4] have proposed an algorithm for stably generating Pad6 approximants. As a by-product, their procedure can also be used to invert general Hankel matrices.…”

Section: Discussionmentioning

confidence: 99%

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

Freund

Zha

1993

Numer. Math.

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Summary. The solution of systems of linear equations with Hankel coefficient matrices can be computed with only O(n 2) arithmetic operations, as compared to O (n 3) operations for the general case. However, the classical Hankel solvers require the nonsingularity of all leading principal submatrices of the Hankel matrix. The known extensions of these algorithms to general Hankel systems can handle only exactly singular submatrices, but not ill-conditioned ones, and hence they are numerically unstable. In this paper, a stable procedure for solving general nonsingular Hankel systems is presented, using a look-ahead technique to skip over singular or ill-conditioned submatrices. The proposed approach is based on a lookahead variant of the nonsymmetric Lanczos process that was recently developed by Freund, Gutknecht, and Nachtigal. We first derive a somewhat more general formulation of this look-ahead Lanczos algorithm in terms of formally orthogonal polynomials, which then yields the look-ahead Hankel solver as a special case. We prove some general properties of the resulting look-ahead algorithm for formally orthogonal polynomials. These results are then utilized in the implementation of the Hankel solver. We report some numerical experiments for Hankel systems with ill-conditioned submatrices. Mathematics Subject Classification (1991): 65F05, 15A57, 42C05

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“…In some exact as well as numeric algorithms, ω is selected as a random p-th root of unity. Then by the theory of early termination of that algorithm [18], the first t × t leading principle submatrix H [t] in the (infinite) Hankel matrix H with entries in row i and column j, Cabay and Meleshko gave an algorithm to bound from above the condition numbers of all the principal submatrices of the associated Hankel system in the well-conditioned case [8], which has been further refined and extended, for example, see [1,6]. Here we focus less on stably computing the Gohberg-Semencul updates (see those cited papers-we require no look-ahead), but on fast and accurate lower and upper bounds of the condition numbers, which constitute the early termination criterion.…”

Section: Introductionmentioning

confidence: 99%

Fast estimates of Hankel matrix condition numbers and numeric sparse interpolation

Kaltofen

Lee

Yang

2012

Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation

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We investigate our early termination criterion for sparse polynomial interpolation when substantial noise is present in the values of the polynomial. Our criterion in the exact case uses Monte Carlo randomization which introduces a second source of error. We harness the Gohberg-Semencul formula for the inverse of a Hankel matrix to compute estimates for the structured condition numbers of all arising Hankel matrices in quadratic arithmetic time overall, and explain how false ill-conditionedness can arise from our randomizations. Finally, we demonstrate by experiments that our condition number estimates lead to a viable termination criterion for polynomials with about 20 non-zero terms and of degree about 100, even in the presence of noise of relative magnitude 10 −5 .

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A Weakly Stable Algorithm for Padé Approximants and the Inversion of Hankel Matrices

Cited by 49 publications

References 46 publications

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

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

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

Fast estimates of Hankel matrix condition numbers and numeric sparse interpolation

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