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

Emre Mengi, Michael L. Overton
2005 IMA Journal of Numerical Analysis  
Two useful measures of the robust stability of the discrete-time dynamical system x k+1 = Ax k 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
more » ... lgorithm, 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. and for real > 0, the -pseudospectrum is The site contains detailed information about pseudospectra, including a comprehensive bibliography and software links. Both the field of values and the -pseudospectrum of A are compact sets containing the eigenvalues of A. Thus, it makes sense to talk about the points in these sets that are located furthest away from the origin. Both the numerical radius of A
doi:10.1093/imanum/dri012 fatcat:bnyvlkgrmna55h7w6jwo7tihiy