解非对称块三对角线性方程组的并行算法

An Efficient Parallel Algorithm for Solving Nonsymmetric Block-Tridiagonal Linear Equations

提出了一种并行求解非对称块三对角线性方程组的方法。该方法通过对传统的预处理共轭梯度法的预条件子进行重新构造,使之适合并行计算。该算法只需相邻两台机子间通信,降低了通信次数易于求解。并从理论上分析文中算法的收敛性,给出了该算法的收敛性优于Gauss-seidel的预处理共轭梯度法的充分条件。最后,在HP rx2600集群上,进行了数值试验,结果表明实算与理论是一致的,并行性好,且迭代次数也明显降低。

Sections 1,2 and 3 of the full paper explain our algorithm,which we believe is more efficient than that of Ref.3;we also believe that it is better in that the mathematics is more complete and rigorous.Section 1 starts from Ref.3 and gives a better construction of the preconditioner of the traditional precondition conjugate gradient method.The better construction is suitable for parallel computing;our algorithm only needs the communication between two adjacent processors,thus reducing the amounts of time of communication.Section 3 analyzes the convergence of our algorithm in theory and obtains a sufficient condition for the algorithm;the convergence is superior to the preconditioners of the block Jacobi precondition matrix and the preceprocessing method of the block Gauss-Seidel precondition matrix;what are particularly worth paying attention to are Theorems 3.2 and 3.4.Section 4 carries out the numerical simulation of our algorithm on the HP rx2600 cluster,the simulation results,given in Tables 1 and 2,and their analysis demonstrate preliminarily that our algorithm can do efficient parallel computing and solve large nonsymmetrical block-tridiagonal linear equations.