邊值問題離散方程組的Gauss—Seidel迭代法的多種存儲格式實現(xiàn)

摘   要

本論文由兩部分組成，第1部分針對邊值問題，用5點差分格式進行離散，并對離散矩陣這類大型稀疏矩陣，研究了系數(shù)矩陣的3種存儲格式的優(yōu)劣，即：滿矩陣存儲格式、半帶寬存儲格式和按行壓縮稀疏存儲格式，首先我們將滿矩陣存儲方式和半帶寬存儲格式進行了對比， 迭代法的數(shù)值實驗表明：利用半帶寬存儲的矩陣在空間運算方面具有高效性；然后針對目前數(shù)值實驗中流行的按行壓縮稀疏存儲格式，實現(xiàn)了有限元離散代數(shù)系統(tǒng)的 迭代法的求解。
在論文的第2部分，我們比較了在3種存儲格式下的 迭代法， 迭代法和 迭代法的優(yōu)劣。最后，作為演示我們將 迭代法， 迭代法和 迭代法用1個例題進行了比較，數(shù)值實驗表明， 迭代法和 迭代法比 迭代法更有效，而超松弛迭代法更優(yōu)。
關(guān)鍵詞：滿矩陣；半帶寬；按行壓縮稀疏； 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 .

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