  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 $n\to\infty$ of the respective inclusion
set to the corresponding spectral set.
