A reflection on the implicitly restarted Arnoldi method for computing eigenvalues near a vertical line
- 作者：Karl Meerbergen ; karl.meerbergen@cs.kuleuven.be ; ; Raf Vandebril raf.vandebril@cs.kuleuven.be
- 关键词：Lyapunov eigenvalue problem ; Kronecker eigenvalue problem ; Eigenvalues closest to the imaginary axis ; Implicitly restarted Arnoldi
- 刊名：Linear Algebra and its Applications
- 出版年：2012
- 期刊代码：38_00243795
- 类别：mt
- 出版时间：15 April, 2012
- 卷：436
- 期：8
- 页码：2828-2844
- 文件大小：363 K
摘要
In this article, we will study the link between a method for computing eigenvalues closest to the imaginary axis and the implicitly restarted Arnoldi method. The extension to computing eigenvalues closest to a vertical line is straightforward, by incorporating a shift. Without loss of generality we will restrict ourselves here to computing eigenvalues closest to the imaginary axis.
In a recent publication, Meerbergen and Spence discussed a new approach for detecting purely imaginary eigenvalues corresponding to Hopf bifurcations, which is of interest for the stability of dynamical systems. The novel method is based on inverse iteration (inverse power method) applied on a Lyapunov-like eigenvalue problem. To reduce the computational overhead significantly a projection was added.
This method can also be used for computing eigenvalues of a matrix pencil near a vertical line in the complex plane. We will prove in this paper that the combination of inverse iteration with the projection step is equivalent to Sorensen鈥檚 implicitly restarted Arnoldi method utilizing well-chosen shifts.