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

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