  We consider sweeping domain decomposition preconditioners to solve the
Helmholtz equation in the case of stripwise domain decomposition with or
without overlaps. We unify their derivation and convergence studies by
expressing them as Jacobi, Gauss-Seidel, and Symmetric Gauss-Seidel methods for
different numbering of the unknowns. The proposed framework enables theoretical
comparisons between the double sweep methods in [Nataf and Nier (1997), Vion
and Geuzaine (2018)] and those in [Stolk (2013, 2017), Vion and Geuzaine
(2014)]. Additionally, it facilitates the introduction of a new sweeping
algorithm. We provide numerical test cases to assess the validity of the
theoretical studies.
