Linear and Multilinear Algebra ( IF 1.1 ) Pub Date : 2020-07-21 , DOI: 10.1080/03081087.2020.1795057 Vaibhav Shekhar 1 , Chinmay Kumar Giri 2 , Debasisha Mishra 1
ABSTRACT
Iterative methods based on matrix splittings are useful tools in solving real large sparse linear systems. In this aspect, the type I double splitting approaches are straight forward from the formulation of the iteration scheme and its convergence theory is well established in the literature. However, if a double splitting is of type II, then the convergence of the iteration scheme seems not to be straight forward. In this paper, we develop convergence theory for type II double splittings to make the implementation quite simple. In this direction, we first introduce two new subclasses of double splittings and establish their convergence theory. Using this theory, we prove a new characterization of a monotone matrix. Finally, we apply our theoretical findings to the double splitting of an M-matrix in the Gauss–Seidel double SOR method to obtain a comparison result.
中文翻译:
II型双弱分裂的一个注记
摘要
基于矩阵分裂的迭代方法是解决实际大型稀疏线性系统的有用工具。在这方面,I 型双分裂方法直接从迭代方案的公式化,其收敛理论在文献中得到了很好的确立。然而,如果双重分裂是类型 II,那么迭代方案的收敛似乎不是直截了当的。在本文中,我们开发了 II 型双分裂的收敛理论,以使实现变得非常简单。在这个方向上,我们首先介绍了双重分裂的两个新子类,并建立了它们的收敛理论。利用这一理论,我们证明了单调矩阵的新表征。最后,我们将我们的理论发现应用于M的双重分裂Gauss-Seidel 双 SOR 方法中的 -matrix 以获得比较结果。