The condition number of real Vandermonde, Krylov and positive definite Hankel matrices |
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Authors: | Bernhard Beckermann |
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Affiliation: | (1) Laboratoire d'Analyse Numa'erique et d'Optimisation, UFR IEEA – M3, UST Lille, F-59655 Villeneuve d'Ascq CEDEX, France; e-mail: bbecker@ano.univ-lille1.fr , FR |
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Abstract: | Summary. We show that the Euclidean condition number of any positive definite Hankel matrix of order may be bounded from below by with , and that this bound may be improved at most by a factor . Similar estimates are given for the class of real Vandermonde matrices, the class of row-scaled real Vandermonde matrices, and the class of Krylov matrices with Hermitian argument. Improved bounds are derived for the case where the abscissae or eigenvalues are included in a given real interval. Our findings confirm that all such matrices – including for instance the famous Hilbert matrix – are ill-conditioned already for “moderate” order. As application, we describe implications of our results for the numerical condition of various tasks in Numerical Analysis such as polynomial and rational i nterpolation at real nodes, determination of real roots of polynomials, computation of coefficients of orthogonal polynomials, or the iterative solution of linear systems of equations. Received December 1, 1997 / Revised version received February 25, 1999 / Published online 16 March 2000 |
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Keywords: | Mathematics Subject Classification (1991):15A12 65F35 |
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