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

## TL;DR

This paper establishes bounds on the spectral condition number of Vandermonde matrices with nearly-colliding pairs of nodes on the unit circle, showing linear growth with inverse separation, and provides sharp constants independent of the total number of nodes.

## Contribution

It offers new bounds and sharp constants for the condition number of Vandermonde matrices with nearly-colliding pairs, extending understanding to this specific collision scenario.

## Key findings

- Condition number grows linearly with inverse node separation.
- Bounds are independent of total node count for well-separated non-colliding nodes.
- Provides sharp constants for pairs of nearly-colliding nodes.

## Abstract

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.