Regularity radius: Properties, approximation and a not a priori exponential algorithm

TL;DR

This paper investigates the regularity radius of matrices, providing new properties, bounds, and algorithms—including a not a priori exponential method—for efficient estimation and computation, especially for special matrix classes.

Contribution

It introduces new properties and bounds for the regularity radius, and presents a polynomial-time algorithm for certain matrix classes, improving upon existing NP-hardness results.

Findings

Checking finiteness of regularity radius is polynomial.

New bounds for regularity radius based on matrix norms.

A not a priori exponential algorithm for regularity testing.

Abstract

The radius of regularity sometimes spelled as the radius of nonsingularity is a measure providing the distance of a given matrix to the nearest singular one. Despite its possible application strength this measure is still far from being handled in an efficient way also due to findings of Poljak and Rohn providing proof that checking this property is NP-hard for a general matrix. To handle this we can either find approximation algorithms or making known bounds for radius of regularity tighter. Improvements of both have been recently shown by Hartman and Hladik (doi:10.1007/978-3-319-31769-4\_9) utilizing relaxation to semidefinite programming. These approaches consider general matrices without or with just mild assumptions about the original matrix. This work explores a process of regularity radius analysis and identifies useful properties enabling easier estimation of the corresponding radius values based on utilization of properties of special class of considered matrices. At first, checking finiteness of regularity radius is shown to be a polynomial problem along with determining a maximal bound on number of nonzero elements of the matrix to obtain infinite radius. Further, relationship between maximum (Chebyshev) norm and spectral norm is used to construct new bounds for the radius of regularity. A new method based on Jansson-Rohn algorithm for testing regularity of an interval matrix is presented which is not a priory exponential along with numerical experiments. For a situation where an input matrix has a special form, several results are provided such as exact formulas for several special classes of matrices, e.g., for totally positive and inverse non-negative, or approximation algorithms, e.g., rank-one radius matrices. For tridiagonal matrices, an algorithm by Bar-On, Codenotti and Leoncini is utilized to design a polynomial algorithm to compute the regularity radius.