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Some problems on the Gauss-Seidel iteration method in degenerate cases. (Chinese. English summary) Zbl 1449.65052
Summary: The Gauss-Seidel iteration method is a highly popular classical iteration algorithm for solving linear systems of equations. It has a profound impact on the development of numerical linear algebra and numerical optimization. In this paper, we mainly discuss the Gauss-Seidel iteration method for solving linear systems of equations associated with self-adjoint and positive semidefinite, but not necessarily positive definite, coefficient operators (i.e., the degenerate case). We provide a review on the development of the convergence analysis for the Gauss-Seidel method, and discuss the related block coordinate descent method applied to the equivalent unconstrained quadratic programming problems. As a consequence, we derive the convergence of the Gauss-Seidel iteration method for the linear equations we considered in this paper. We also compare the convergence analysis and results of the Gauss-Seidel iteration with the symmetric Gauss-Seidel iteration. The differences observed from this comparison not only motivate the proof provided in this paper, but also pave the way for related research topics in the future. Finally, we highlight some unresolved questions that are highly related to this paper and leave them as future research topics.
65F10 Iterative numerical methods for linear systems