光子晶體所衍生之高維度零空間特徵值問題在平移Lanczos下的影響

Abstract

為了利用數值方法去探討三維光子晶體的能帶結構，我們需要解由Maxwell方程組所決定的特徵值問題，除非找到一個合適的解特徵值的方法才能有效率地計算特徵值問題，利用Yee’s scheme離散方法得到此特徵值問題，我們提出使用Shift-and-invert Lanczos方法和Shifted Lanczos方法來解決由此產生的特徵值問題。此外，我們從ARPACK中改寫的Lanczos方法在Lanczos過程中有較佳的計算量。

To explore band structures of three-dimensional photonic crystals numerically, we need to solve the eigenvalue problems derived from the governing Maxwell's equations. The solutions of these eigenvalue problems cannot be computed efficiently unless a suitable eigensolver. Taking eigenvalue problems due to Yee's scheme as examples, we propose using Shift-and-invert Lanczos method and shifted Lanczos method to solve the resulting eigenvalue problems. We rewrite the Lanczos method from ARPACK and find it better calculations in the Lanczos process.