Convergent Spectral Inclusion Sets for Banded Matrices.

We obtain sequences of inclusion sets for the spectrum, essential spectrum, and pseudospectrum of banded, in general non-normal, matrices of finite or infinite size. Each inclusion set is the union of the pseudospectra of certain submatrices of a chosen size 𝑛. Via the choice of 𝑛, one can balance accuracy of approximation against computational cost, and we show, in the case of infinite matrices, convergence as 𝑛→∞ of the respective inclusion set to the corresponding spectral set.