尋找對稱矩陣的極特徵根之疊代方法

Abstract

在很多科學的領域，例如物理和化學，有時必須對角化大型的對稱矩陣。最常用來找一些極特徵根的方法是Davidson和Jacobi-Davidson方法。在這篇論文中，我們同時提出和測試一個命名為『sweep』的方法，它是一個用來找對稱矩陣極特徵根的好工具。在為數不少的帶狀矩陣中，sweep方法比Davidson和Jacobi-Davidson方法表現來的好。失去擴充向量的正交化是一個從這些方法延伸出來的重大問題。我們發現解決它的方式是正交化兩次。在未來，我們需要用更多不同形式的矩陣來驗證sweep方法的效能。

In many scientific fields like physics and chemistry one often has to diagonalize large symmetric matrices. The two most popular methods of finding extreme eigenvalues of such large matrices are Davidson and Jacobi-Davidson methods. In this thesis we propose and test a new method, called “sweep method”, that is an efficient tool for finding extreme eigenvalues of large symmetric matrices. We have found that it has better performance than Davidson and Jacobi-Davidson methods for a large class of band matrices. A serious numerical problem obeserved for all these methods is the loss of orthogonality among the expansion vectors. We have found tha a way to avoid this problem is doing two reorthogonalizations. In the future, we need more different types of matrices to confirm the efficiency of the sweep method.