An implicitly restarted rational Krylov strategy for Lyapunov inverse iteration
Karl Meerbergen · IMA Journal of Numerical Analysis · 2015
The computation of eigenvalues nearest the imaginary axis is a hard problem. It is a useful tool for computing eigenvalues with largest real part (also called right-most eigenvalues) of matrix pairs arising from the stability analysis of a dynamical system. We present an efficient implementation of the Lyapunov inverse iteration method, presented by Meerbergen and Spence [(2010) Shift-and-invert iteration for purely imaginary eigenvalues with application to the detection of Hopf bifurcations in large scale problems. SIAM J. Matrix Anal. Appl., 31, 1463–1482]. It turns out that when we use the approximate power Lyapunov solver, the method corresponds to the implicitly restarted rational Krylov method and resembles the Iterative Rational Krylov Algorithm for model reduction. Elman and Wu [(2013) Lyapunov inverse iteration for computing a few right-most eigenvalues of large generalized eigenvalue problems. SIAM J. Matrix Anal. Appl., 34, 1685–1707] proved that an accurate solution of the Lyapunov equation guarantees the accurate computation of eigenvalues, and often, the right-most eigenvalues. However, the approach in this paper is usually cheaper in terms of memory and allows us to compute more than one eigenvalue more efficiently.