On the numerical condition of a generalized Hankel eigenvalue problem

The generalized eigenvalue problem ̃ Hy= λHy with H a Hankel matrix and ̃ 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. Mathematics Subject Classification (2000) 15A18 · 65F35 · 15A12 · 30E10 B. Beckermann was supported in part by INTAS research network NaCCA 03-51-6637. G. H. Golub was supported in part by DOE grant DE-FC-02-01ER41177. G. Labahn was supported in part by NSERC and MITACS Canada grants. B. Beckermann (B) Laboratoire Painlevé UMR 8524 (ANO-EDP), UFR Mathématiques – M3, UST Lille, 59655 Villeneuve d’Ascq Cedex, France e-mail: [email protected] G. H. Golub Fletcher Jones Professor of Computer Science, Stanford University, Stanford, CA, USA e-mail: [email protected] G. Labahn David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada e-mail: [email protected] 42 B. Beckermann et al.