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The condition number of real Vandermonde, Krylov and positive definite Hankel matrices. (English) Zbl 0965.15003

Every real semipositive definitive Hankel matrix has the form \(H_{n} (\mu):=[ h_{i+j+1}] _{i,j=0,1,\dots,n}\) where \(h_{k}:=\int x^{k}d\mu(x)\) for some positive Borel measure \(\mu\) defined on the real line. We get the special case of the Hilbert matrix when \(\mu\) is the uniform measure on \([0,1]\). For each interval \(I\subseteq\mathbb{R}\) the author defines \(\Gamma_{n}(I)^{2}\) to be the infimum of the Euclidean condition numbers \(\kappa_{2}(H_{n}(\mu))\) as \(\mu\) ranges over measures with \(\text{supp}(\mu)\subseteq I\).
This paper derives good upper and lower bounds on \(\Gamma_{n}(I)\) of the form \(a_{I}\leq\Gamma_{n} (I)(\lambda_{I})^{-n}\leq b_{I}\) (one must square these bounds to get bounds on condition numbers!). The following special cases give a flavour of the results. For \(I=[0,1]\), \(\lambda_{I}=(1+\sqrt{2})^{2}\approx 5.828\), \(a_{I}=(2\sqrt{2n+2})^{-1}\) and \(b_{I}=\sqrt{2n+1};\) in comparison it is known that the Hilbert matrix has a condition number which is asymptotic to a constant times \((1+\sqrt{2})^{4n}/\sqrt{n}\) [see H. S. Wilf, “Finite sections of some classical inequalities”, Springer (1970; Zbl 0199.38301), Eq. (3.35)] so the general bound is quite close. On the other hand, for \(I=\mathbb{R}\), \(\lambda_{I}=\exp(2\cdot Catalan/\pi)\approx 1.792\), \(a_{I}=(4\sqrt {n+1})^{-1}\) and \(b_{I}=\lambda_{I}/\sqrt{2}\) which shows that every positive definite Hankel matrix has condition number \(\geq(3.210)^{n}/16n.\)
The paper ends with a discussion of some applications to the condition numbers of Vandermonde matrices and the sensitivity of polynomial roots, and to the problem of rational interpolation.
Reviewer: J.D.Dixon (Ottawa)

MSC:

15A12 Conditioning of matrices
65F35 Numerical computation of matrix norms, conditioning, scaling
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory

Citations:

Zbl 0199.38301
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