  Recently Ahmadi et al. (2021) and Tagliaferro (2022) proposed some iterative
methods for the numerical solution of linear systems which, under the classical
hypothesis of strict diagonal dominance, typically converge faster than the
Jacobi method, but slower than the forward/backward Gauss-Seidel one. In this
paper we introduce a general class of iterative methods, based on suitable
splittings of the matrix that defines the system, which include all of the
methods mentioned above and have the same cost per iteration in a sequential
computation environment. We also introduce a partial order relation in the set
of the splittings and, partly theoretically and partly on the basis of a
certain number of examples, we show that such partial order is typically
connected to the speed of convergence of the corresponding methods. We pay
particular attention to the case of linear systems for which the Jacobi
iteration matrix is nonnegative, in which case we give a rigorous proof of the
correspondence between the partial order relation and the magnitude of the
spectral radius of the iteration matrices. Within the considered general class,
some new specific promising methods are proposed as well.
