On the exponential rate of the condition number of Fourier submatrices and Vandermonde matrices

TL;DR

This paper determines the precise exponential growth rate of the condition number for Fourier and Vandermonde submatrices, revealing fundamental limits on their numerical stability.

Contribution

It provides the exact exponential rate of ill-conditioning for specific Fourier and Vandermonde submatrices, including tight bounds involving Catalan's constant.

Findings

Exact exponential ill-conditioning rate for square submatrices with contiguous rows and columns.

A tight upper bound of 2G/π on the exponential rate for all submatrices with contiguous columns.

General analysis of Vandermonde-like matrices with explicit estimates based on logarithmic potentials.

Abstract

The discrete Fourier transform matrix is one of the most important matrices in linear algebra, and submatrices of it arise in a variety of applications. Though the discrete Fourier transform matrix is unitary, its submatrices can be exponentially ill-conditioned, an obstacle to accurate computation. This work resolves the exact rate of the exponential ill-conditioning for square submatrices with contiguous rows and columns. As a consequence, we obtain a tight upper bound of $2 G/\pi$ on the exponential rate for all submatrices with contiguous columns, or, equivalently, all Vandermonde submatrices with distinct support points, where $G$ is Catalan's constant. These results follow from a more general analysis of Vandermonde and Vandermonde-like matrices for which exact estimates for exponential ill-conditioning are developed in terms of logarithmic potentials.