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The Scientific World Journal
Volume 2014, Article ID 273873, 10 pages
Research Article

Convergence Results on Iteration Algorithms to Linear Systems

1School of Mathematical Science, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China
2Department of Mathematics, Zhejiang Ocean University, Zhoushan, Zhejiang 316000, China
3School of Information Science and Engineering, East China University of Science and Technology, Shanghai 200237, China

Received 15 April 2014; Accepted 21 April 2014; Published 13 May 2014

Academic Editor: Shan Zhao

Copyright © 2014 Zhuande Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In order to solve the large scale linear systems, backward and Jacobi iteration algorithms are employed. The convergence is the most important issue. In this paper, a unified backward iterative matrix is proposed. It shows that some well-known iterative algorithms can be deduced with it. The most important result is that the convergence results have been proved. Firstly, the spectral radius of the Jacobi iterative matrix is positive and the one of backward iterative matrix is strongly positive (lager than a positive constant). Secondly, the mentioned two iterations have the same convergence results (convergence or divergence simultaneously). Finally, some numerical experiments show that the proposed algorithms are correct and have the merit of backward methods.