  The Douglas-Rachford algorithm is a classical and very successful method for
solving optimization and feasibility problems. In this paper, we provide novel
conditions sufficient for finite convergence in the context of convex
feasibility problems. Our analysis builds upon, and considerably extends,
pioneering work by Spingarn. Specifically, we obtain finite convergence in the
presence of Slater's condition in the affine-polyhedral and in a
hyperplanar-epigraphical case. Various examples illustrate our results.
Numerical experiments demonstrate the competitiveness of the Douglas-Rachford
algorithm for solving linear equations with a positivity constraint when
compared to the method of alternating projections and the method of
reflection-projection.
