Differential Equations for Roaming Pseudospectra: Paths to Extremal Points and Boundary Tracking

When studying the $\varepsilon$-pseudospectrum of a matrix, one is often interested in computing the extremal points having maximum real part or modulus. This is a crucial step, for example, when computing the distance to instability of a stable system. Using the property that the pseudospectrum is determined via perturbations by rank-1 matrices, we derive differential equations on the manifold of normalized rank-1 matrices whose solutions tend to the critical rank-1 perturbations associated with the extremal points of (locally) maximum real part and modulus. This approach also allows us to track the boundary contour of the pseudospectrum in a neighborhood of the extremal points. The technique we propose is related to an idea recently developed by Guglielmi and Overton, who derived discrete dynamical systems instead of the continuous ones we present. The method turns out to be fast in comparison with those previously proposed in the literature and appears to be promising in dealing with large sparse problems.