An Experimental Comparison of Methods for Computing the Numerical Radius

TL;DR

This paper experimentally compares different computational methods for determining the numerical radius of complex matrices, focusing on accuracy and efficiency of optimization-based approaches.

Contribution

It provides an empirical evaluation of two main methods based on different mathematical characterizations, highlighting their practical performance.

Findings

Convex optimization method is more accurate but computationally intensive.

Nonconvex optimization method is faster but less precise.

Results guide choosing appropriate methods based on accuracy and speed needs.

Abstract

We make an experimental comparison of methods for computing the numerical radius of an $n\times n$ complex matrix, based on two well-known characterizations, the first a nonconvex optimization problem in one real variable and the second a convex optimization problem in $n^{2}+1$ real variables. We make comparisons with respect to both accuracy and computation time using publicly available software.