Inverting matrices of pairwise minima

Let 0<x1<x2<…<xn and let A be the symmetric n×n matrix with ijth entry Aij=min.¹ This matrix has linearly independent columns and so is invertible. Its inverse A^{-1} is symmetric, tridiagonal, and has {ij}^\text{th} entry [A^{-1}]_{ij}=\begin{cases} \frac{1}{x_1}+\frac{1}{x_2-x_1} & \text{if}\ i=j=1 \\ \frac{1}{x_i-x_{i-1}}+\frac{1}{x_{i+1}-x_i} & \text{if}\ 1<i=j<n \\ \frac{1}{x_n-x_{n-1}} & \text{if}\ i=j=n \\ -\frac{1}{x_j-x_i} & \text{if}\ i=j-1 \\ -\frac{1}{x_i-x_j} & \text{if}\ i=j+1 \\ 0 & \text{otherwise}. \end{cases} For example, if x_i=2^{i-1} for each i\le n=5 then A=\begin{bmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 2 & 2 \\ 1 & 2 & 4 & 4 & 4 \\ 1 & 2 & 4 & 8 & 8 \\ 1 & 2 & 4 & 8 & 16 \\ \end{bmatrix} and A^{-1}=\begin{bmatrix} 2 & -1 & 0 & 0 & 0 \\ -1 & 1.5 & -0.5 & 0 & 0 \\ 0 & -0.5 & 0.75 & -0.25 & 0 \\ 0 & 0 & -0.25 & 0.375 & -0.125 \\ 0 & 0 & 0 & -0.125 & 0.125 \\ \end{bmatrix} You may wonder: why is this useful? Suppose I observe data \{(x_i,y_i)\}_{i=1}^n, where the function f:[0,\infty)\to\mathbb{R} mapping regressors x_i\ge0 to outcomes y_i=f(x_i) is the realization of a Wiener process. I use these data to estimate some value f(x) via Bayesian regression. My estimate depends on the inverse of the covariance matrix for the outcome vector y=(y_1,y_2,\ldots,y_n). This matrix has {ij}^\text{th} entry \min\{x_i,x_j\}, so I can compute its inverse using the expression above.