  The well-known Asplund theorem states that the inverse of a (possibly
one-sided) band matrix $A$ is a Green matrix. In accordance with quasiseparable
theory, such a matrix admits a quasiseparable representation in its
rank-structured part. Based on this idea, we derive algorithms that compute a
quasiseparable representation of $A^{-1}$ with linear complexity. Many
inversion algorithms for band matrices exist in the literature. However,
algorithms based on a computation of the rank structure performed theoretically
via the Asplund theorem appear for the first time in this paper. Numerical
experiments confirm complexity estimates and offer insight into stability
properties.

Numerical Analysis;Numerical Analysis