# Vandermonde Matrices with Nodes in the Unit Disk and the Large Sieve

## TL;DR

This paper establishes new bounds on the singular values and condition number of Vandermonde matrices with nodes in the unit disk, linking them to large sieve inequalities and improving upon previous bounds with better stability and simplicity.

## Contribution

It introduces a novel connection between Vandermonde matrices with nodes in the unit disk and large sieve inequalities, providing sharper and more numerically stable bounds than prior results.

## Key findings

- Derived sharper bounds on condition numbers of Vandermonde matrices.
- Established a new link between singular values and large sieve inequalities.
- Improved upon existing bounds for nodes on the unit circle.

## Abstract

We derive bounds on the extremal singular values and the condition number of NxK, with N>=K, Vandermonde matrices with nodes in the unit disk. The mathematical techniques we develop to prove our main results are inspired by a link---first established by by Selberg [1] and later extended by Moitra [2]---between the extremal singular values of Vandermonde matrices with nodes on the unit circle and large sieve inequalities. Our main conceptual contribution lies in establishing a connection between the extremal singular values of Vandermonde matrices with nodes in the unit disk and a novel large sieve inequality involving polynomials in z \in C with |z|<=1. Compared to Baz\'an's upper bound on the condition number [3], which, to the best of our knowledge, constitutes the only analytical result---available in the literature---on the condition number of Vandermonde matrices with nodes in the unit disk, our bound not only takes a much simpler form, but is also sharper for certain node configurations. Moreover, the bound we obtain can be evaluated consistently in a numerically stable fashion, whereas the evaluation of Baz\'an's bound requires the solution of a linear system of equations which has the same condition number as the Vandermonde matrix under consideration and can therefore lead to numerical instability in practice. As a byproduct, our result---when particularized to the case of nodes on the unit circle---slightly improves upon the Selberg-Moitra bound.