The paper introduces a general class of iterative splitting methods for solving linear systems of the form Ax = b, where A is the coefficient matrix. The key idea is to split the Jacobi iteration matrix BJ = L + U into multiple parts, and perform the iteration in successive steps using these parts.
The main highlights and insights are:
The authors define the concept of a splitting mask M(d) and the corresponding splitting B(d) of the Jacobi iteration matrix BJ. This allows them to formulate a general iterative scheme (2.2) that encompasses various existing methods as special cases.
The authors prove the cyclicity property of the splittings, showing that cyclic shifts of a splitting lead to spectrum-equivalent iteration matrices.
The authors introduce a partial order relation of "refinement" on the set of splittings, and show that more refined splittings typically lead to faster convergence, especially when the Jacobi iteration matrix BJ is nonnegative.
The authors extend the classical convergence results under strict diagonal dominance to the whole class of splitting methods proposed.
The authors show that the iterative methods proposed by Ahmadi et al. (2021) and Tagliaferro (2022) are special cases of the general class introduced here.
The authors propose some new specific splitting methods, such as the "alternate triangular column/row methods", which seem to have good potential for fast convergence.
Overall, the paper provides a unifying framework for analyzing and developing efficient iterative splitting methods for solving large linear systems.
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