很多應用問題需要求解大量大型加邊線性系統，例如拓延法中的預測與校正過程。若原未加邊線性系統已具有高效率的算法，則塊狀消去法是為利用此優點以有效求解其加邊線性系統。Govaerts et al[4,5,6]針對以Crout與Doolittle塊狀分解所推導出來的塊狀消去法進行研究與分析，並結合這兩個演算法的優點提出了一個穩定且有效率的混合塊狀消去法求解加一邊的小型線性系統。本文章將探討是否可將其結果推廣到求解多邊的大型線性系統。最後，我們將推廣的結果應用在求解非線性薛丁格方程中。

Many applications need to solve a number of large bordered linear systems such as the prediction and correction processes in continuation method.If the original linear system already has a highly e cient solver, then the block elimination algorithms are designed for solving its bordered linear system e ciently by taking this advantage. Govaerts et al[4, 5, 6]study the block elimination algorithms derived from Crout and Doolittle block factorizations. Combining the advantages of two algorithms, they propose a stable and ef cient mixed block elimination algorithm for solving small bordered linear systems with one border. In this paper we will study whether their results can be extended to solve large bordered linear systems with more borders. In the end we will apply the extended results in solving nonlinear Schrodinger equations.

[1.論文簡介+page1]
[2.塊狀消去法+page2]
[2.1 Crout的塊狀消去法(BEC)+page2]
[2.1.1 BEC演算法+page3]
[2.2 Doolittle的塊狀消去法(BED)+page3]
[2.2.1 BED演算法+page4]
[2.3 改良演算法+page4]
[2.3.1 Alg+k演算法+page5]
[3.塊狀分解演算法數值實驗+page5]
[3.1 數值結果+page6]
[4.混合塊狀消去法(BEM)+page7]
[4.1 BEM演算法+page7]
[4.1.1 數值結果+page8]
[4.2 二層加邊矩陣+page8]
[ 4.2.1 數值結果+page9]
[ 4.3 三層加邊矩陣+page11]
[4.3.1 數值結果+page11]
[4.4 K層加邊矩陣+page13]
[4.4.1 k=10+page13]
[4.4.2 k=20+page14]
[4.4.3 k=20+page16]
[5.連續法中的應用+page18]
[5.1 離散化+page19]
[5.1.1 二維拉普拉斯算子離散化+page19]
[5.1.2 角動量算子離散化+page21]
[5.1.3 非線性薛丁格方程式及能量泛函離散化+page23]
[5.2 追蹤曲線+page23]
[5.3 截取曲線+page24]
[5.4 BEM演算法應用+page25]
[5.4.1 一層加邊矩陣+page26]
[5.4.2 二層加邊矩陣+page26]
[5.4.3 誤差分析+page28]
[6.結論+page31]
[7.參考文獻+page32]

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