Multivariate Vandermonde matrices with separated nodes on the unit circle are stable

TL;DR

This paper establishes explicit bounds on the stability of multivariate Vandermonde matrices with nodes on the unit circle, showing how separation affects their condition number across dimensions.

Contribution

It provides the first explicit bounds on the smallest singular value and condition number for these matrices, improving previous results and analyzing the effect of node separation and dimension.

Findings

Condition number is uniformly bounded when node separation grows linearly with dimension.

Condition number grows slightly faster than exponential when separation grows logarithmically.

Results are quasi optimal and improve upon all previous bounds.

Abstract

We prove explicit lower bounds for the smallest singular value and upper bounds for the condition number of rectangular, multivariate Vandermonde matrices with scattered nodes on the complex unit circle. Analogously to the Shannon-Nyquist criterion, the nodes are assumed to be separated by a constant divided by the used polynomial degree. If this constant grows linearly with the spatial dimension, the condition number is uniformly bounded. If it grows only logarithmically with the spatial dimension, the condition number grows slightly stronger than exponentially with the spatial dimension. Both results are quasi optimal and improve over all previously known results of such type.