Low-Rank Dynamics for Computing Extremal Points of Real Pseudospectra

We consider the real $\varepsilon$-pseudospectrum of a real square matrix, which is the set of eigenvalues of all real matrices that are $\varepsilon$-close to the given matrix, where closeness is measured in either the 2-norm or the Frobenius norm. We characterize extremal points and compare the situation with that for the complex $\varepsilon$-pseudospectrum. We present differential equations for rank-1 and rank-2 matrices for the computation of the real pseudospectral abscissa and radius. Discretizations of the differential equations yield algorithms that are fast and well suited for sparse large matrices. Based on these low-rank differential equations, we further obtain an algorithm for drawing boundary sections of the real pseudospectrum with respect to both the 2-norm and the Frobenius norm.

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