On ε-spectra and stability radii

Techniques of Krylov subspace iterations play an important role in computing ε-spectra of large matrices. To obtain results about the reliability of this kind of approximations, we propose to compare the position of the ε-spectrum of A with those of its diagonal submatrices. We give theoretical results which are valid for any block decomposition in four blocks, A11,A12,A21,A22. We then illustrate our results by numerical experiments. The same kind of problem arises when we compute the stability radius of a large matrix. In that context, we propose a new sufficient condition for the stability of a matrix involving quantities readily computable such as stability radius of small submatrices.