用Krylov子空間迭代演算法來求解Rayleigh商數為對稱矩陣的特徵值

Optimal algorithms in a Krylov subspace for iteratively solving the Rayleigh quotient as the eigenvalue of a symmetric matrix

林君諺(Jin-Yen Lin)

指導教授 ： 劉進賢

國立臺灣大學/工學院/土木工程學研究所/碩士(2014年)

https://doi.org/10.6342/NTU.2014.03000

摘要

特徵值問題是許多工程及數學領域上很常見的問題，也有許多其他形式的問題經轉換而成特徵值問題，如何有效精確地求解特徵值問題是工程師很重要的課題。
本研究想藉由已經存在的Krylov 子空間法及Rayleigh quotient(雷利商數)的概念並且配合Arnoldi化過程來延伸出新的迭代方法，去解決矩陣的特徵值問題然而迭代的過程不只一種方式，算例中會加以比較選出最好的方法並且應用在各類型的矩陣中，而土木工程界中大型結構的振態模數也可以用此方法來探討。
本研究的方法可以算出matlab所無法跑出的大型對稱矩陣的特徵值，譬如到 以上的大型矩陣，故可以解決此大型矩陣無法運算的問題。
最後此方法只能用在對稱矩陣中而非對稱矩陣並無法拿來求解，原因會在最後的結論中提到。

關鍵字

特徵值 ； Krylov 子空間法 ； Rayleigh quotient(雷利商數) ； Arnoldi化過程 ； 振態模數

並列摘要

Eigenvalue problem is a very common problem in engineering and mathematics, there are many other forms of problem converted from eigenvalue problem, and how to efficiently and accurately solve the eigenvalue problem is a very important topic for engineerning.
This study wants to present Krylov subspace methods and Rayleigh quotient concept and with the Arnoldi process to extend the new iterative method to solve the matrix eigenvalue problem, however iterative process more than one way, this study will compare and select the best method applied in various types of matrix, and mode shapes modulus of large structures in civil engineering can also use this method to explore the efficiency of new method.
The method of this study can calculated eigenvales where matlab can not run out of inlarge symmetric matrices.
Finally, this method can only be used in a symmetric matrix symmetric matrix, the reason will be mentioned in the final conclusion.

並列關鍵字

Eigenvalue ； Krylov subspace ； Rayleigh quotient ； Arnoldi process ； mode shapes

參考文獻

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