用块上三角分裂法求解由大不定最小二乘问题引起的块三乘三线性系统

IF 3.4 2区 数学 Q1 MATHEMATICS, APPLIED
Jun Li , Kailiang Xin , Lingsheng Meng
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引用次数: 0

摘要

在本研究中,我们主要利用平稳迭代方法结合Krylov子空间技术,如GMRES,来解决大型不定最小二乘问题。为此,首先将大型不定最小二乘问题的正规方程转化为具有非奇异对角块的稀疏分块3 × 3线性系统,然后对分块3 × 3线性系统的系数矩阵进行分块上三角矩阵的分裂,这种分裂不仅产生平稳迭代方法,而且自然地导出了一个预条件;该方法可以在GMRES方法中用于求解块线性系统。从理论上证明了迭代法具有无条件收敛性。此外,该理论还证明了预条件矩阵的所有特征值都是实数且位于正区间内。最后,数值结果表明理论结果是正确的,所研究的方法也是有效的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A block upper triangular splitting method for solving block three-by-three linear systems arising from the large indefinite least squares problem
In this research, we mainly utilize the stationary iteration method in conjunction with Krylov subspace techniques, such as GMRES, to tackle the large indefinite least squares problem. To accomplish this, the normal equation of the large indefinite least squares problem is firstly transformed into the sparse block three-by-three linear systems with non-singular diagonal blocks, then a block upper triangular matrix splitting of the coefficient matrix of the block three-by-three linear systems is given, the splitting not only produces the stationary iteration method, but also naturally derives a preconditioner, which can be used within GMRES method to solve the block linear systems. Thereafter, it is proved theoretically that the iteration method has unconditional convergence. Furthermore, the theory also shows that all the eigenvalues of the preconditioned matrix are real number and located in a positive interval. In the end, numerical results reflect that the theoretical results are correct and the studied methods are also effective.
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来源期刊
CiteScore
7.90
自引率
10.00%
发文量
755
审稿时长
36 days
期刊介绍: Applied Mathematics and Computation addresses work at the interface between applied mathematics, numerical computation, and applications of systems – oriented ideas to the physical, biological, social, and behavioral sciences, and emphasizes papers of a computational nature focusing on new algorithms, their analysis and numerical results. In addition to presenting research papers, Applied Mathematics and Computation publishes review articles and single–topics issues.
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