I am trying to find the inverse of the following symmetric positive definite matrix : $$ \left(\begin{array}{6*c} 4& 1&0 & &\cdots&0\\ 1& 4& 1& &\huge0& \vdots \\ 0& \ddots& \ddots& \ddots& & \\ \vdots && \ddots& \ddots& \ddots&0\\ & \huge 0 & & 1& 4& 1\\ 0& \cdots& &0& 1& 4 \end{array}\right) $$
Of course the mathematical inverse of this matrix is full, however numerically I observed that the coefficients' magnitude of the inverse matrix are decreasing at an exponential rate around the diagonal.
Far from the border effect (i.e. for $i$ sufficiently greater than $1$ and lower than $n$) the diagonal pattern is the same up to $10^{-20}$: ($\forall j, i \neq i', M^{-1}_{i'j} \approx M^{-1}_{ij} $). Moreover, I empirically check that this behavior does not depend on the dimension of the linear system $n$ (same pattern).
Do you have any clue on how to justify this observation? Or better, an analytical form to compute this inverse?
Cheers,
Leo.
The explicit formulas for inversion of tridiagonal matrices can be found in many places, e.g., https://link.springer.com/article/10.1007/BF01396436
A tridiagonal Toeplitz is just a special case. See for example in https://core.ac.uk/download/pdf/144007569.pdf page 15.