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Title: 退化情形下高斯-赛德尔迭代法的几个问题
Other Title: Some problems on the Gauss-Seidel iteration method in degenerate cases
Authors: Chen, L 
Sun, D 
Toh, KC
Issue Date: 2019
Source: 数值计算与计算机应用 (Journal of numerical methods and computer applications), 2019, v. 40, no. 1, p. 98-110
Abstract: 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 GaussSeidel 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 will 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.
Keywords: Gauss-Seidel iteration
Symmetric Gauss-Seidel iteration
Linear system of equations
Unconstrained convex quadratic programming
Block coordinate descent
Publisher: 科学出版社
Journal: 数值计算与计算机应用 (Journal of numerical methods and computer applications) 
ISSN: 1000-3266
Rights: © 2019 China Academic Journal Electronic Publishing House. It is to be used strictly for educational and research use.
© 2019 中国学术期刊电子杂志出版社。本内容的使用仅限于教育、科研之目的。
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