LOW-RANK DYNAMICS FOR COMPUTING EXTREMAL POINTS OF REAL PSEUDOSPECTRA NICOLA GUGLIELMI ∗ AND CHRISTIAN LUBICH ‡ Abstract. We consider the real ε-pseudospectrum of a real square matrix, which is the set of eigenvalues of all real matrices that are ε-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 ε-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. Key words. Real pseudospectrum, real pseudospectral abscissa, real stability radius. AMS subject classifications. 15A18, 65K05 1. Introduction. In this paper a novel approach to computing extremal points of real pseudospectra is presented and analyzed. We are interested in computing effects of perturbations on the spectrum of a given real matrix A. The framework consists of (i) fixing a class of perturbations (here we consider the class of real and the class of complex perturbations); (ii) fixing a norm to measure the size of the possible perturbations (here we consider the Frobenius and the spectral norm). We measure the largest possible spectral abscissa or radius of the perturbed spec- trum, or - in other words - the pseudospectral abscissa or radius. In the literature these quantities are widely studied since they allow to analyze stability properties and robustness of stable linear dynamical systems; see, e.g., [HP05]). The most common tool is the unstructured complex pseudospectrum, which means that independently of the structure of the matrix, unstructured perturbations of a given norm are con- sidered. When one restricts the analysis to structured perturbations, challenging mathematical difficulties arise, because the characterization in terms of singular val- ues of the standard unstructured pseudospectrum is lost when we put constraints on the perturbations. In this paper we focus our attention on real perturbations of real matrices, but in an appendix we consider also the case of complex perturbations for comparison. In particular we are able to exploit a low-rank property of critical perturbations, which allows us to devise efficient methods to compute the extremal points. In the literature very few papers discuss real pseudospectra (for example [BRQ98] and [Ru06]) and their computation appears to be difficult. For a square matrix A, let Λ(A) denote its spectrum. Let now K = R or C, and let the norm ‖·‖ be the 2-norm ‖·‖ 2 or the Frobenius norm ‖·‖ F on K n×n . For real ε> 0, the ε-pseudospectrum of a matrix A ∈ K n×n is given by Λ K,‖·‖ ε (A)= {λ ∈ C : λ ∈ Λ(A + E) for some E ∈ K n×n with ‖E‖≤ ε}. (1.1) ∗ Dipartimento di Matematica Pura ed Applicata, Universit`a degli Studi di L’ Aquila, Via Vetoio - Loc. Coppito, I-67010 L’ Aquila, Italy. Email: guglielm@univaq.it ‡ Mathematisches Institut, Universit¨at T¨ ubingen, Auf der Morgenstelle 10, D–72076 T¨ ubingen, Germany. Email: lubich@na.uni-tuebingen.de 1