雅可比-戴維森和反位移變換殘量阿洛迪方法的使用說明書

Abstract

計算大型矩陣的特徵值問題一直是很困難的問題，使用疊代法可以有效的解決。

本文中我們應用疊代投影法中的雅可比-戴維森(Jacobi-Davidson) 和反位移變換殘量阿洛迪(Shift-Invert Residual Arnoldi) 方法，利用不精確疊代，把原問題投影到子空間中縮小矩陣的維度，計算近似的特徵對。

我們提供兩個程式jdsiraSEP 和jdsiraGEP，這兩個程式是由黃建智博士使用MATLAB 語言撰寫，它們分別解決大型標準特徵值問題和廣義特徵值問題，並加入重啟動(restarting ) 和鎖住(locking) 技巧，增加計算效率。使用者可選擇擴大子空間的方法雅可比-戴維森或反位移變換殘量阿洛迪，計算出需要的特徵對。

在本文中我們介紹jdsiraSEP 和jdsiraGEP 的演算法及如何在MATLAB 上操作，最後提供一個實際的範例光子晶體演示如何解決特徵值問題。

Solving large scale eigenvalue problems is always very difficult, it is efficient by using the Iteration method to solve this problem. In this thesis, we mean the iterative projection methods, Jacobi-Davidson (JD) and Shift-Invert Residual Arnoldi (SIRA) methods, which using inexact iteration. The problem be projection onto subspace, in order to reduce dimension, then we can obtain approximate by solve the smaller eigenvalue problem.

We provide two package, jdsiraSEP and jdsiraGEP, which were primarily written by Chien-Chih Huang via MATLAB, these can solve the standard eigenvalue problems and generalized eigenvalue problems, respectively.

The jdsiraSEP and jdsiraGEP based on Jacobi-Davidson and Shift-Invert Residual Arnoldi methods are applied to restarting and locking techniques. User can choose JD or SIRA method to find the desired eigenpairs.

We then describe the algorithm about the jdsiraSEP andjdsiraGEP and how to operate in MATLAB. At the end of this thesis, we take the photonic crystals for example to show how to solve the eigenproblem.