Algorithms for the computation of the pseudospectral radius and the numerical radius of a matrix

Two useful measures of the robust stability of the discrete-time dynamical system xk+1 = Axk are the -pseudospectral radius and the numerical radius of A. The -pseudospectral radius of A is the largest of the moduli of the points in the -pseudospectrum of A, while the numerical radius is the largest of the moduli of the points in the field of values. We present globally convergent algorithms for computing the -pseudospectral radius and the numerical radius. For the former algorithm, we discuss conditions under which it is quadratically convergent and provide a detailed accuracy analysis giving conditions under which the algorithm is backward stable. The algorithms are inspired by methods of Byers, Boyd– Balakrishnan, He–Watson and Burke–Lewis–Overton for related problems and depend on computing eigenvalues of symplectic pencils and Hamiltonian matrices.