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Lower bounds for the condition number of Vandermonde matrices. (English) Zbl 0646.15003

Let \(V_ n(x)\) be an \(n\times n\) Vandermonde matrix, \(\kappa_{n,\infty}(x)=\| V_ n(x)\|_{\infty}\| V_ n^{- 1}(x)\|_{\infty}\) and \(\kappa_{n,\infty}=\inf \kappa_{n,\infty}(x)\) taken over all nonnegative nodes \(x_ 1>x_ 2>...>x_ n\geq 0\). Then \(\kappa_{n,\infty}\geq (n-1)\{1+(1-n^{- 1})^{-1/(n-1)}\}^{n-1}\) for \(n\geq 2\). Similar results are obtained for nodes located symmetrically with respect to the origin. The paper is based in part on earlier work of the first author [ibid. 24, 1-12 (1975; Zbl 0316.65005)]. To assess the quality of those bounds, numerical work on the spectral conditions number is included.
Reviewer: E.Kreyszig

MSC:

15A12 Conditioning of matrices
15A42 Inequalities involving eigenvalues and eigenvectors
65F35 Numerical computation of matrix norms, conditioning, scaling

Citations:

Zbl 0316.65005
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References:

[1] Garbow, B.S., Boyle, J.M., Dongarra, J.J., Moler, C.B.: Matrix Eigensystem Routines-EISPACK Guide Extension. Lecture Notes in Computer Science, Vol. 51. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0368.65020
[2] Gautschi, W.: On Inverses of Vandermonde and Confluent Vandermonde Matrices. Numer. Math.4, 117-123 (1962) · Zbl 0108.12501 · doi:10.1007/BF01386302
[3] Gautschi, W.: Norm Estimates for Inverses of Vandermonde Matrices. Numer. Math.23, 337-347 (1975) · Zbl 0304.65031 · doi:10.1007/BF01438260
[4] Gautschi, W.: Optimally Conditioned Vandermonde Matrices. Numer. Math.24, 1-12 (1975) · Zbl 0316.65005 · doi:10.1007/BF01437212
[5] Gautschi, W.: On Inverses of Vandermonde and Confluent Vandermonde Matrices III. Numer. Math.29, 445-450 (1978) · Zbl 0362.15002 · doi:10.1007/BF01432880
[6] Golub, G.H., Van Loan, C.F.: Matrix Computations. Baltimore: Johns Hopkins University Press 1983
[7] IMSL Library Reference Manual4, Edition 9, 1982
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