带状线性方程组的含参交替方向并行算法

Parallel Alternating Direction Algorithm with Parameters for Solving Banded Linear Systems

在MIMD分布式存储环境下针对系数矩阵为带状或块三对角矩阵的线性方程组提出了含三参数交替方向迭代并行算法。通过引入三参数调整,并适当分裂系数矩阵得到新算法,给出了系数矩阵为若干特殊矩阵时算法的收敛条件。在HP rx2600集群系统上实现了算法,针对不同的算例将其与多分裂方法、BSOR方法和PEk内迭代方法进行了比较。并行计算结果表明,所提算法具有较高的加速比和并行效率,明显优于多分裂方法和PEk方法,能合理分配内存,从而有效节省计算时间。针对算例1,加速比和计算效率略优于BSOR方法;而算例2的结果明显优于PEk内迭代方法。

This paper focused on parallel iterative method with parameters for solving banded or block tridiagonal linear systems on distributed-memory cluster. By splitting the coefficient matrix and using parameters, we proposed a new al- gorithm and gave some convergence theories for some special coefficient matrices. Furthermore, we implemented the al- gorithm on HP rx2600 cluster and compared it with multisplitting method, BSOR method and PEk inner iterative me- thod for different examples. The numerical experiments indicate that acceleration rates and efficiency of our algorithm are higher than the multi-splitting one. The algorithm saves computational time by allocating memory properly. As to Example 1, the acceleration rates and efficiency of our algorithm are better than the BSOR one slightly. And the results for Example 2 are better than PEk inner iterative one significantly.