ISSN:
0945-3245
Keywords:
Mathematics Subject Classification (1991):15A12, 65F35
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/PL00005392
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