边值问题离散方程组的Gauss—Seidel迭代法的多种存储格式实现

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翻新时间：2023-07-18

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边值问题离散方程组的Gauss—Seidel迭代法的多种存储格式实现

摘要

本论文由两部分组成，第1部分针对边值问题，用5点差分格式进行离散，并对离散矩阵这类大型稀疏矩阵，研究了系数矩阵的3种存储格式的优劣，即：满矩阵存储格式、半带宽存储格式和按行压缩稀疏存储格式，首先我们将满矩阵存储方式和半带宽存储格式进行了对比，迭代法的数值实验表明：利用半带宽存储的矩阵在空间运算方面具有高效性；然后针对目前数值实验中流行的按行压缩稀疏存储格式，实现了有限元离散代数系统的迭代法的求解。

在论文的第2部分，我们比较了在3种存储格式下的迭代法，迭代法和迭代法的优劣。最后，作为演示我们将迭代法，迭代法和迭代法用1个例题进行了比较，数值实验表明，迭代法和迭代法比迭代法更有效，而超松弛迭代法更优。

关键词：满矩阵；半带宽；按行压缩稀疏；迭代法；超松弛迭代法。

Abstract

This thesis consists of two parts. The first one was that boundary value problem was discrete with five point difference method. Three kinds of memory formats were studied, which were full matrix, half band width and row compress sparse. Full matrix memory format and half band width memory format first was contrasted. The results show that half band width has efficiency in space. Then the algebraic system of finite element method was solved by Gauss-Seidel iteration method to popular row compress sparse memory format.

In the second one, Jacobi method, Gauss-Seidel method and Successive Over-Relaxation (SOR) method were compared under the three kinds of memory formats. In the end, an example was used to demonstrate. Results indicate that Gauss-Seidel method and SOR method are move valid than Jacobi method, and SOR method is the best one.

Keywords: Full matrix; Half band width; row compress sparse memory; Gauss-Seidel iteration method; SOR iteration method .

说明：论文中有些数学符号是编辑器编辑而成，网页上无法显示或者显示格式错误，给您带来不便请谅解。