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標題: Lanczos方法解延續問題
Lanczos type Methods for Continuation Problems
作者: 翁振陵
關鍵字: Lanczos方法
Lanczos methods
Continuation problems
Eigenvalue problems
Conjugate gradient methods
出版社: 應用數學研究所
摘要: 我們研究Lanczos型態的方法處理分支問題。首先,我們指出如何利用對稱的Lanczos方法解正定與不定的聯立方程式,而且它可用來偵測特徵值問題分支解上的簡單分支點,這包含計算係數矩陣的部分三角化矩陣的最小特徵值、最小奇異值、及其條件數;其次,塊狀的Lanczos方法可用來偵測多重分支點及解多重右邊向量的聯立方程式,我們亦嘗試用非對稱的Lanczos方法可精確地估計一個接近對稱的矩陣之最小特徵值,因而方可用來偵測簡單分支點。我們亦討論到加前置條件的技巧;最後對抽樣的數值結果提出報告,我們的試驗問題函蓋了:階的半線性特徵值問題與四階的von Karman方程。
We study Lanczos type methods for continuation problems. First we indicate how the symmetric Lanczos method may be used to solve both positive definite linear systems and indefinite systems. Furthermore, it can be used to monitor the simple bifurcation points on the solution curve of the eigenvalue problems. This includes computing the minimum eigenvalue, the minimum singular value, and the condition number of the partial tridiagonalization of the coefficient matrix. The Ritz vector thus obtained can be applied to compute the tangent vector at the bifurcation point for branch-switching. Next, we indicate that the block Lanczos method can be used to monitor the multiple bifureations as well as to solve the multiple right hand sides. We also show that the unsymmetric Lanczos method can be exploited to compute the minimum eigenvalue of a near symmetric matrix, and therefore to detect the simple bifurcation point as well. Some preconditioning techniques are discussed. Sample numerical results are reported. Our test problems include second order semilinear elliptic eigenvalue problems and von Karman equations.
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