State space realizations of rational interpolants with prescribed poles

Ribalta, Angel

doi:10.1016/s0167-6911(01)00082-2

Cited by 3 publications

(1 citation statement)

References 7 publications

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“…They naturally arise when computing rational interpolates with prescribed poles [28]. This type of interpolation has applications in control systems [30]. Cauchy-Vandermonde matrices also appear in connection with the numerical solution of singular integral equations [9,18], as well as in numerical quadrature [31] and rational models of regression and E-optimal design [14,17].…”

Section: Introductionmentioning

confidence: 99%

Accurate bidiagonal decomposition of totally positive Cauchy–Vandermonde matrices and applications

Marco

Martínez

Peña

2017

Linear Algebra and its Applications

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Cauchy-Vandermonde matrices play a fundamental role in rational interpolation theory and in other fields. When all their corresponding nodes are different and positive and all poles are different and negative and follow adequate orderings, these matrices are totally positive. In this paper we provide fast algorithms for computing bidiagonal factorizations of these matrices and their inverses with high relative accuracy. These algorithms can be used to solve with high relative accuracy other algebraic problems, such as the computation of all singular values, all eigenvalues or the solution of certain linear systems. The error analysis of the algorithm for computing the bidiagonal factorization and the corresponding perturbation theory are also performed.

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Section: Introductionmentioning

confidence: 99%

Accurate bidiagonal decomposition of totally positive Cauchy–Vandermonde matrices and applications

Marco

Martínez

Peña

2017

Linear Algebra and its Applications

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Accurate bidiagonal decompositions of Cauchy–Vandermonde matrices of any rank

Delgado,

Koev,

Marco

et al. 2024

Numerical Linear Algebra App

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We present a new decomposition of a Cauchy–Vandermonde matrix as a product of bidiagonal matrices which, unlike its existing bidiagonal decompositions, is now valid for a matrix of any rank. The new decompositions are insusceptible to the phenomenon known as subtractive cancellation in floating point arithmetic and are thus computable to high relative accuracy. In turn, other accurate matrix computations are also possible with these matrices, such as eigenvalue computation amongst others.

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Accurate solutions of weighted least squares problems associated with rank-structured matrices

Huang

2019

Applied Numerical Mathematics

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State space realizations of rational interpolants with prescribed poles

Cited by 3 publications

References 7 publications

Accurate bidiagonal decomposition of totally positive Cauchy–Vandermonde matrices and applications

Accurate bidiagonal decomposition of totally positive Cauchy–Vandermonde matrices and applications

Accurate bidiagonal decompositions of Cauchy–Vandermonde matrices of any rank

Accurate solutions of weighted least squares problems associated with rank-structured matrices

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