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Lower Bounds for the Condition Number of a Real Confluent Vandermonde Matrix

Ren-Cang Li
Mathematics of Computation
Vol. 75, No. 256 (Oct., 2006), pp. 1987-1995
Stable URL: http://www.jstor.org/stable/4100134
Page Count: 9
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Lower Bounds for the Condition Number of a Real Confluent Vandermonde Matrix
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Abstract

Lower bounds on the condition number $\kappa_{p}(V_c)$ of a real confluent Vandermonde matrix Vc 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 kmax (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 kmax.

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