Parallelization of Iterative Method

采用迭代法对线性系统进行了数值计算。在本课题中，我们提出了一种新的预处理迭代方法，并通过一些数值实验证明了我们的方法比SOR方法更快。我们的方法是高斯-塞德尔方法，用于预处理具有正参数的矩阵。并推导了该方法的收敛定理。此外，我们还得到了预条件矩阵最优参数的估计公式。进一步对该方法的有效性和对流扩散问题进行了验证，结果表明该方法对解决这类实际问题是有效的。众所周知，如果系数矩阵A有h矩阵，则迭代法可以用于求解线性系统。因此，我们开发了一种简单的判别h矩阵的先验方法。在科学和工程的各个领域中，我们所提出的预处理迭代法对于提高求解线性系统的性能是有效的。

The numerical computation was carried out to solve the linear system by using the iterative method.In this project we presented the new preconditioning iterative method and showed that our method is faster than SOR method with some numerical experiments. Our method is the Gauss-Seidel Method for the preconditioning matrix with a positive parameter. We also derived convergence theorem of our proposed method. Moreover, we obtained the estimation formula of optimum parameter for the preconditioned matrix. Further we tested the effectivity of the proposed method and convective diffusion problem, and we obtained that our method is effective in solving such a practical problem.It is well known that if coefficient matrix A has H-matrix, then the iterative method is able to use solving the linear systems. Thus, we developed simple a priori method for judging H-matrix.It has been pointed out that our preconditioning iterative method is effective to improve the performance to solve the linear system in various fields of science and engineering.