A reflection on the implicitly restarted Arnoldi method for computing eigenvalues near a vertical line

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 eigenvalues closest to a vertical line is straightforward, by incorporating a shift. Without loss of generality we will restrict ourselves here to the imaginary axis. In a recent publication, Meerbergen and Spence discussed a new approach for detecting purely imaginary eigenvalues corresponding to Hopf bifurcations. The proposed method is based on inverse iteration (inverse power method) on a Lyapunov like eigenvalue problem. A projection step was added that significantly reduces the computational overhead. The same method can be used for computing eigenvalues of a matrix pencil near a vertical line in the complex plane. This method then appears to be equivalent with Sorensen’s implicitly restarted Arnoldi method with a special choice of shifts.