A Bisection Method for Measuring the Distance of a Stable Matrix to the Unstable Matrices

Abstract

We describe a bisection method to determine the 2-norm and Frobenius norm distances from a given matrix A to the nearest matrix with an eigenvalue on the imaginary axis. If A is stable in the sense that its eigenvalues lie in the open-left half plane, then this distance measures how “nearly unstable“ A is. Each step provides either a rigorous upper bound or a rigorous lower bound on the distance. A few bisection steps can bracket the distance within an order of magnitude. Bisection avoids the difficulties associated with nonlinear minimization techniques and the occasional failures associated with heuristic estimates. We show how the method might be used to estimate the distance to the nearest matrix with an eigenvalue on the unit circle.