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

Abstract:

Lower bounds on the condition number $\kappa _p(V_{\mathrm {c}})$ of a real confluent Vandermonde matrix $V_{\mathrm {c}}$ 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), $\kappa _p(V_{\mathrm {c}})$ 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 $k_{\max }$.