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.
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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.
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Beckermann, B., Golub, G.H. & Labahn, G. On the numerical condition of a generalized Hankel eigenvalue problem. Numer. Math. 106, 41–68 (2007). https://doi.org/10.1007/s00211-006-0054-x
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DOI: https://doi.org/10.1007/s00211-006-0054-x