  Analysis of pipe networks involves computing flow rates and pressure
differences on pipe segments in the network, given the external inflow/outflow
values. This analysis can be conducted using iterative methods, among which the
algorithms of Hardy Cross and Newton-Raphson have historically been applied in
practice. In this note, we address the mathematical analysis of the local
convergence of these algorithms. The loop-based Newton-Raphson algorithm
converges quadratically fast, and we provide estimates for its convergence
radius to correct some estimates in the previous literature. In contrast, we
show that the convergence of the Hardy Cross algorithm is only linear. This
provides theoretical confirmation of experimental observations reported earlier
in the literature.
