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 x(k+1) = Ax(k) are the epsilon-pseudospectral radius and the numerical radius of A. The epsilon-pseudospectral radius of A is the largest of the moduli of the points in the epsilon-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 epsilon-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.