- 學位論文
特殊五對角線矩陣線性系統之相關探討
Research on Toeplitz Pentadiagonal linear system
摘要
在電腦上做一些特殊的計算時，常會用帶寬矩陣線性系統的方式來表現一連串的計算式，而在解這問題時，計算速度的關鍵就在於矩陣線性方程式上，隨著原始問題的不同，這個矩陣可能是三對角線或五對角線甚至更多，我們便想要改善這運算的複雜度以求有更好的計算效率。 近年來關於三對角線矩陣線性系統的研究已經接近完善，所以我們往五對角線矩陣線性系統來做研究，我們先考察了近年來關於特殊五對角線矩陣線性系統的相關研究，從這些研究中我們可以了解更多有關特殊五對角線線性系統的性質。 在這篇論文中，首先我們先說明何謂五對角線矩陣以及五對角線線性系統，並且為了表示其實用性，我們提供了一個計算中使用到特殊五對角線矩陣線性系統的問題。接著我們選出兩個解特殊五對角線線性系統的方法，並且說明其計算方法以得知其優缺點。 最後我們針對Toeplitz五對角線矩陣來做研究，想辦法加速其運算。在參考三對角線系統的方法後，我們想以類似的方法來加速五對角線矩陣系統的運算，利用LR分解的兩個矩陣在電腦計算表示上會有收斂的特性，進而減少不必要的重複運算。由實驗結果可以證明此方法是可行的，以此法可以以較少的運算達到近似高斯消去法的結果，而達到加速的效果。我們也用此方法解之前舉出的問題，而證實其實用性。
並列摘要
When we solve some problems on computer calculation, we can convert the series of computing to the banded linear system. Then the key of the computing speed is the computing of matrix linear system. Along with the problems, the banded matrix may be tridiagonal or pentadiagonal or more. We try to improve the computing speed of the matrix linear system to obtain better efficiency of solving the problem. Because the works of tridiagonal linear system were well enough, we take the focus on pentadiagonal linear system. Then we surveyed some recent topics about pentadiagonal matrix. With these survey, we can get more understandings about the pentadiagonal linear system. In this thesis, We show what pentadiagonal matrix is and pentadiagonal system of linear equation First. In order to show the practicality of the pentadiagonal system, we choose an example using pentadiagonal system of linear equation to solve the problem. Second, we select two methods that could solve the special kind of pentadiagonal linear equation system, and introduce these methods to see the merits and demerits of them. Finally we focus on the performance of solving pentadiagonal Toeplitz matrix. In this work, we propose the new approach for solving real symmetric pentadiagonal Toeplitz matrix systems of linear equations. Our algorithm entails fewer floating-point operations when it be compared with Gaussian elimination method. According to our experiment result, our result can be solve the shallow water problem faster than Gaussian elimination.