I have a band matrix -- a sparse, square, symmetric $N \times N$ matrix whose structure looks like the following:
Here, the area under the blue stripes is the non-zero elements; everything else is zero
Is there an algorithm to invert this kind of matrix that is simple yet more efficient than Gaussian elimination and LU decomposition?
Since none of the comments gave the concrete answer, I'll write it explicitly here in case anyone needs it (like I did).
Firstly, unfortunately, the inverse of a band-limited matrix is a full (non-band-limited) matrix in general, so just filling out the entries of the inverse matrix would take $\Omega\left(n^2\right)$. So I'll assume you just want to solve a linear system $A x = b$.
Using the algorithm in this paper, a general band-limited matrix $A$ of size $n \times n$ with bandwidth $k$ can be decomposed into triangular $k$-bandwidth matrices $L$ and $U$ in $O \left( k^2 n \right)$ time. From there, $L U x = b$ can be solved quickly in $O(k n)$ time. So overall, the runtime will be $O\left(k^2 n\right)$. As a followup, if $k$ is constant, that means that the system can be solved in linear time (highly useful).
