Momentum accelerated power iterations and the restarted Lanczos method

Abstract

In this paper we compare two methods for finding extremal eigenvalues and eigenvectors: the restarted Lanczos method and momentum accelerated power iterations. The convergence of both methods is based on ratios of Chebyshev polynomials evaluated at subdominant and dominant eigenvalues; however, the convergence is not the same. Here we compare the theoretical convergence properties of both methods, and determine the relative regimes where each is more efficient. We further introduce a preconditioning technique for the restarted Lanczos method using momentum accelerated power iterations, and demonstrate its effectiveness. The theoretical results are backed up by numerical tests on benchmark problems.