Consider a symmetric positive definite tridiagonal linear system $$A x = b$$ where $A \in \mathbb{R}^{n \times n}$ and $b \in \mathbb{R}^n$. Given three indices $0 \le i < j < k < n$, if we assume only equation rows strictly between $i$ and $k$ hold, we can eliminate intermediate variables to get an equation of the form $$u x_i + v x_j + w x_k = c$$ where $v > 0$. This equation relates the value of $x_j$ to $x_i,x_k$ independent of "outside" influence (say, if a constraint affecting $x_0$ was introduced).
Question: Is it possible to preprocess the linear system $Ax = b$ in $O(n)$ time so that the linking equation for any $(i,j,k)$ can be determined in $O(1)$ time?
If the diagonal of $A$ is 2, the offdiagonals are $-1$, and $b = 0$, the desired result is the analytic result for the discretized Poisson equation. Unfortunately, it is not possible to transform a general SPD tridiagonal system into a constant coefficient Poisson equation without breaking the tridiagonal structure, essentially because different variables can have different levels of "screening" (locally strict positive definiteness). A simple diagonal scaling of $x$, for example, can eliminate half of the $2n-1$ DOFs of $A$ but not the other half.
Intuitively, a solution to this problem would require arranging the problem so that the amount of screening could be accumulated into a linear size array and then somehow "cancelled" to arrive at the linking equation for the given triple.
Update (more intuition): In terms of PDEs, I have a discretized linear elliptic problem in 1D, and I want to know whether I can spend $O(n)$ in precomputation to produce some sort of "analytic" solution that can be looked up in $O(1)$ time, where I am allowed to vary where the boundary conditions are.