A curve tracing algorithm for computing the pseudospectrum

The boundary curve of the pseudospectrum of a matrix is defined as a contour line of its resolvent norm. A rather simple and efficient continuation method is presented, which determines the implicitly given curve by a prediction-correction scheme, where the correction step is accomplished by one single Newton step. Besides its efficiency the algorithm turns out to be very accurate as long as the boundary of the pseudospectrum is a smooth curve. Problems may arise at bifurcation points where the resolvent norm is not differentiable.This work was partially supported by Deutsche Forschungsgemeinschaft.