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

Summary.

We show that the Euclidean condition number of any positive definite Hankel matrix of order $n\geq 3$ may be bounded from below by $\gamma^{n-1}/(16n)$ with $\gamma=\exp(4 \cdot{\it Catalan}/\pi) \approx 3.210$, and that this bound may be improved at most by a factor $8 \gamma n$. 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 i nterpolation at real nodes, determination of real roots of polynomials, computation of coefficients of orthogonal polynomials, or the iterative solution of linear systems of equations.