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Mathematical Problems in Engineering
Volume 2014, Article ID 752651, 8 pages
Research Article

Parallel Algorithm with Parameters Based on Alternating Direction for Solving Banded Linear Systems

1Department of Applied Mathematics, Xidian University, Xi’an 710071, China
2Department of Applied Mathematics, Xianyang Normal University, Xianyang 712000, China
3Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, China
4School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710072, China

Received 3 December 2013; Accepted 30 January 2014; Published 7 April 2014

Academic Editor: Massimo Scalia

Copyright © 2014 Xinrong Ma 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.


An efficient parallel iterative method with parameters on distributed-memory multicomputer is investigated for solving the banded linear equations in this work. The parallel algorithm at each iterative step is executed using alternating direction by splitting the coefficient matrix and using parameters properly. Only it twice requires the communications of the algorithm between the adjacent processors, so this method has high parallel efficiency. Some convergence theorems for different coefficient matrices are given, such as a Hermite positive definite matrix or an -matrix. Numerical experiments implemented on HP rx2600 cluster verify that our algorithm has the advantages over the multisplitting one of high efficiency and low memory space, which has a considerable advantage in CPU-times costs over the BSOR one. The efficiency for Example 1 is better than BSOR one significantly. As to Example 2, the acceleration rates and efficiency of our algorithm are better than the PEk inner iterative one.