On the numerical condition of a generalized Hankel eigenvalue problem

Authors:
B. Beckermann;G. H. Golub;G. Labahn
Affiliations:
UFR Mathématiques – M3, Laboratoire Painlevé UMR 8524 (ANO-EDP), UST Lille, 59655, Villeneuve d’Ascq Cedex, France;Stanford University, Fletcher Jones Professor of Computer Science, UST Lille, 59655, Stanford, CA, USA;University of Waterloo, David R. Cheriton School of Computer Science, UST Lille, 59655, Waterloo, ON, Canada
Venue:
Numerische Mathematik
Year:
2007

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Abstract

The generalized eigenvalue problem $$\widetilde H y \,{=}\, \lambda H y$$ with H a Hankel matrix and $$\widetilde H$$ the corresponding shifted Hankel matrix occurs in number of applications such as the reconstruction of the shape of a polygon from its moments, the determination of abscissa of quadrature formulas, of poles of Padé approximants, or of the unknown powers of a sparse black box polynomial in computer algebra. In many of these applications, the entries of the Hankel matrix are only known up to a certain precision. We study the sensitivity of the nonlinear application mapping the vector of Hankel entries to its generalized eigenvalues. A basic tool in this study is a result on the condition number of Vandermonde matrices with not necessarily real abscissas which are possibly row-scaled.