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arxiv: 1812.08645 · v2 · pith:EHAZFAINnew · submitted 2018-12-20 · 🧮 math.NA · cs.NA

On the condition number of Vandermonde matrices with pairs of nearly-colliding nodes

Stefan Kunis , Dominik Nagel This is my paper
classification 🧮 math.NA cs.NA
keywords nodesnumberconditionpairscollidematricesvandermondebounds
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We prove upper and lower bounds for the spectral condition number of rectangular Vandermonde matrices with nodes on the complex unit circle. The nodes are "off the grid", pairs of nodes nearly collide, and the studied condition number grows linearly with the inverse separation distance. Such growth rates are known in greater generality if all nodes collide or for groups of colliding nodes. For pairs of nodes, we provide reasonable sharp constants that are independent of the number of nodes as long as non-colliding nodes are well-separated.

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    Proves that the smallest singular value of clustered multivariate Vandermonde matrices on the unit circle is bounded below by the product of inverse intra-cluster distances, with matching upper bounds and sharp consta...