量子點模擬所產生之多項式特徵值問題的數值研究

A numerical study for polynomial eigenvalue problems arising in quantum dot simulations

李昌澤(Chang-Tse Lee)

指導教授 ： 黃聰明

國立臺灣師範大學/理學院/數學系/碩士(2009年)

摘要

在本文中，我們討論了如何使用多項式 Jacobi-Davidson 疊代法去解決多項式特徵值問題。此外，我們使用 locking 的方式把已收斂的值給保留住，並且給予四個 scheme 來解用多項式 Jacobi-Davidson 疊代法解多項式特徵值問題所產生的 correction 方程式。
我們的多項式特徵值問題是來自量子點模擬，其中包含三種不同形狀的量子點和兩種有效質量模型。在數值結果中，我們說明了係數矩陣裡的非零元素在這五個問題中的表現，以及比較各種 schenme 來解 correction 方程式以及使用不同的 precondition 的效果，用以建議一個最佳的選擇在每一個不同的情況下，用來解多項式 Jacobi-Davidson 疊代法解多項式特徵值問題所產生的 correction 方程式。

關鍵字

多項式特徵值問題 ； Jacobi-Davidson 法 ； 薛丁格方程式

並列摘要

In this paper, we study how to use the polynomial Jacobi-Davidson iterative mehtod to solve the polynomial eigenvalues problem. And we use the locking scheme to lock the convergent eigenpaors, and give four schemes to solve the correction equation in polynomial Jacobi-Davidson method. A set of polynomial eigenvalue benchmark problems are derived from quantum dot simulations, with three different shapes of quantum dot and two kinds of effective mass models. In numerical results, we illustrate the non-zero elements of all coefficient matrices in these five benchmark problems and compare the performance of the various schemes for solving correction equation with different preconditioners to suggest the best choice in each different case when solving the correction equation in polynomial Jacobi-Davidson method.

並列關鍵字

polynomial eigenvalue problems ； Jacobi-Davidson method ； Schr¨odinger equation

參考文獻

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