The spectral properties of Vandermonde matrices with clustered nodes

TL;DR

This paper analyzes the spectral properties of Vandermonde matrices with clustered nodes on the unit circle, showing near orthogonality of different clusters and deriving precise estimates for singular values and condition numbers.

Contribution

It introduces a novel spectral analysis approach for Vandermonde matrices with clustered nodes, simplifying the problem by reducing it to individual cluster analysis.

Findings

Pairs of cluster subspaces are nearly orthogonal.

Provides accurate estimates for singular values of the matrix.

Derives componentwise condition numbers for least squares problems.

Abstract

We study rectangular Vandermonde matrices $\mathbf{V}$ with $N+1$ rows and $s$ irregularly spaced nodes on the unit circle, in cases where some of the nodes are "clustered" together -- the elements inside each cluster being separated by at most $h \lesssim {1\over N}$, and the clusters being separated from each other by at least $\theta \gtrsim {1\over N}$. We show that any pair of column subspaces corresponding to two different clusters are nearly orthogonal: the minimal principal angle between them is at most $$\frac{\pi}{2}-\frac{c_1}{N \theta}-c_2 N h,$$ for some constants $c_1,c_2$ depending only on the multiplicities of theclusters. As a result, spectral analysis of $\mathbf{V}_N$ is significantly simplified by reducing the problem to the analysis of each cluster individually. Consequently we derive accurate estimates for 1) all the singular values of $\mathbf{V}$, and 2) componentwise condition numbers for the linear least squares problem. Importantly, these estimates are exponential only in the local cluster multiplicities, while changing at most linearly with $s$.