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Lower bounds for the condition number of a real confluent Vandermonde matrix

Author: Ren-Cang Li
Journal: Math. Comp. 75 (2006), 1987-1995
MSC (2000): Primary 15A12, 65F35
Published electronically: May 16, 2006
MathSciNet review: 2240645
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Abstract: Lower bounds on the condition number $ \kappa_p(V_{{c}})$ of a real confluent Vandermonde matrix $ V_{{c}}$ are established in terms of the dimension $ n$, or $ n$ and the largest absolute value among all nodes that define the confluent Vandermonde matrix and the interval that contains the nodes. In particular, it is proved that for any modest $ k_{\max}$ (the largest multiplicity of distinct nodes), $ \kappa_p(V_{{c}})$ behaves no smaller than $ {\mathcal O}_n((1+\sqrt 2\,)^n)$, or than $ {\mathcal O}_n((1+\sqrt 2\,)^{2n})$ if all nodes are nonnegative. It is not clear whether those bounds are asymptotically sharp for modest $ k_{\max}$.

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Additional Information

Ren-Cang Li
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506

Keywords: Optimal condition number, Vandermonde matrix, confluent Vandermonde matrix, Chebyshev polynomials
Received by editor(s): October 20, 2004
Received by editor(s) in revised form: May 23, 2005
Published electronically: May 16, 2006
Additional Notes: This work was supported in part by the National Science Foundation CAREER award under Grant No. CCR-9875201 and by the National Science Foundation under Grant No. DMS-0510664.
Article copyright: © Copyright 2006 American Mathematical Society