One efficient method (using <A HREF="https://en.wikipedia.org/wiki/Hessenberg_matrix">Hessenberg matrices</A>) is described here: <A HREF="http://faculty.nps.edu/pstanica/research/2013JCAM_BandedMatr.pdf">The inverse of banded matrices</A> (2013).

> Let $B_{r,n}$ ($1 \leq r \leq n$) be an $n \times n$ matrix of entries
> $\{a_{ij}\}$, $−r \leq i \leq r$, $1 \leq j \leq r$, with the
> remaining un-indexed entries all zeros. In this paper we give the LU
> factorization and the inverse of the matrix $B_{r,n}$, using a method 
> based on Hessenberg submatrices associated to $B_{r,n}$.