On the numerical condition of a generalized Hankel eigenvalue problem

The generalized eigenvalue problem with H a Hankel matrix and 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. 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.