Global convergence of the restarted Lanczos and Jacobi–Davidson methods for symmetric eigenvalue problems

Abstract

The Lanczos method is well known for computing the extremal eigenvalues of symmetric matrices. For efficiency and robustness a restart strategy is employed in practice, but this makes an analysis of convergence less straightforward. We prove global convergence of the restarted Lanczos method in exact arithmetic using certain convergence properties of the Rayleigh–Ritz procedure, which can be obtained from the discussion by Crouzeix, Philippe, and Sadkane. For the restarted Lanczos method, Sorensen’s previous analysis establishes global convergence to the largest eigenvalues under the technical assumption that the absolute values of the off-diagonal elements of the Lanczos tridiagonal matrix are larger than a positive constant throughout the iterations. In this paper, we prove global convergence without any such assumption. The only assumption is that the initial vector is not orthogonal to any of the target exact eigenvectors. More importantly, our results are applicable to dynamic restarting procedures where the dimensions of the projection subspace are dynamically determined. In other words, our analysis can be applied to the recently proposed efficient restart strategies employed in the thick restarted Lanczos method. The convergence theorem is extended to the restarted Lanczos method for computing both the largest and smallest eigenvalues. Moreover, we derive certain global convergence theorems of the block Lanczos and Jacobi–Davidson methods, where, for both algorithms, the Ritz values are shown to converge to exact eigenvalues, although they are not necessarily extremal.