Inverting matrices of pairwise minima

Let 0<x1<x2<…<xn and let A be the symmetric n×n matrix with ijth entry Aij=min{xi,xj}.¹ This matrix has linearly independent columns and so is invertible. Its inverse A−1 is symmetric, tridiagonal, and has ijth entry [A−1]ij={1x1+1x2−x1if i=j=11xi−xi−1+1xi+1−xiif 1<i=j<n1xn−xn−1if i=j=n−1xj−xiif i=j−1−1xi−xjif i=j+10otherwise. For example, if xi=2i−1 for each i≤n=5 then A=[11111122221244412488124816] and A−1=[2−1000−11.5−0.5000−0.50.75−0.25000−0.250.375−0.125000−0.1250.125] You may wonder: why is this useful? Suppose I observe data {(xi,yi)}ni=1, where the function f:[0,∞)→R mapping regressors xi≥0 to outcomes yi=f(xi) 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=(y1,y2,…,yn). This matrix has ijth entry min{xi,xj}, so I can compute its inverse using the expression above.