# Optimizing the inverse of a symmetric matrix with 3 non consecutive diagonal terms

Let $$A$$ be a $$200x200$$ matrix that is symmetric and has 3 diagonal terms. It is like a tridiagonal matrix but the diagonals are non-consecutive. Or like a banded matrix with bandwidth 100 but with zeros in between. The shape of the matrix is more clearer when shown using spy(A) in Matlab: Shape of the matrix

I want to compute the inverse in a very efficient way since I am running a code in real-time. Performing the regular inverse (using Eigen3 library in C++) takes around 0.3s and it needs to be less than 0.1s.

$$A^{-1}$$ is also symmetric and has the exact same shape. This leads me to believe that there must be a property or a way to calculate the inverse more efficiently. However, it seems like $$A$$ does not match any standard type of matrix (not banded matrix, not tridiagonal...) apart from being symmetric. The Cholesky decomposition does not help at all.

Does anyone know if there is a standard equation or algorithm or property to calculate $$A^{-1}$$. Thanks so much!