One efficient method (using Hessenberg submatrices) is described here: The inverse of banded matrices (2013).

One efficient method (using Hessenberg matrices) is described here: The inverse of banded matrices (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 $B_{r,n}$.