On the Convergence of an Implicitly Restarted Arnoldi Method
Richard B. Lehoucq · University of North Texas Digital Library (University of North Texas) · 1999
. We show that Sorensen's [35] implicitly restarted Arnoldi method (including its block extension) is non-stationary simultaneous iteration with an implicit projection step to accelerate convergence to the invariant subspace of interest. By using the geometric convergencetheory for nonstationary simultaneous iteration due to Watkins and Elsner [43], we prove that an implicitly restarted Arnoldi method can achieve a super-linear rate of convergence to the dominant invariant subspace of a matrix. Moreover, we show how an IRAM computes a nested sequence of approximations for the partial Schur decomposition associated with the dominant invariant subspace of a matrix. Key words. Simultaneous iteration, Arnoldi reduction, Schur decomposition, restarting, eigenvalues. AMS subject classifications. 65F15, 65G05 1. Introduction. A classical method of solution for the large-scale eigenvalue problem is simultaneous iteration [6, 9, 26, 27, 30, 37, 40]. Simultaneous iteration was originally intr...