  A popular method for finding the projection onto the intersection of two
closed convex subsets in Hilbert space is Dykstra's algorithm. In this paper,
we provide sufficient conditions for Dykstra's algorithm to converge rapidly,
in finitely many steps. We also analyze the behaviour of Dykstra's algorithm
applied to a line and a square. This case study reveals stark similarities to
the method of alternating projections. Moreover, we show that Dykstra's
algorithm may stall for an arbitrarily long time. Finally, we present some open
problems.
