@Matthew Knepley It's all the same for tridiagonal. Yes, we could generalize to banded, in which case we could call it SPIKE -- it's just a matter of what language is used to describe it. Anyway, I have no interest in solving with a single system at a time, but solving many different systems in parallel is useful for directional-split algorithms. CRWENO has the property that the reduced system is so well-conditioned that a direct coarse solve doesn't pay off because the iterative solve is guaranteed to converge in a small number of iterations. Different systems may need a direct coarse solve, in which case you have choices for parallelism (addressed in tridiagLU and the paper).