The condition number of real Vandermonde, Krylov and positive definite Hankel matrices

Abstract

We show that the Euclidean condition number of any positive definite Hankel matrix of order n ≥ 3 may be bounded from below by γn-1/(16n) with γ = exp(4 · Catalan/π) ≈ 3.210, and that this bound may be improved at most by a factor 8γn. 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 interpolation at real nodes, determination of real roots of polynomials, computation of coefficients of orthogonal polynomials, or the iterative solution of linear systems of equations.