Building and solving matrix spectral problems

When an infinite dimensional operator T: X → X is approximated with (a slight perturbation of) an operator T_n : X → X of finite rank less than or equal to n, the spectral elements of an auxiliary matrix Z ∈ ℂ^{n ×n} , lead to those of Tn, if they are computed exactly. This contribution covers a general theoretical framework for matrix problems issued from finite rank discretizations and perturbed variants, the stop criterion of the QR method for eigenvalues, the possibility of using the Newton method to compute a Schur form, and the use of Newton method to refine coarse approximate bases of spectral subspaces. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)