A simple extrapolation method for clustered eigenvalues

Abstract

This paper introduces a simple variant of the power method. It is shown analytically and numerically to accelerate convergence to the dominant eigenvalue/eigenvector pair, and it is particularly effective for problems featuring a small spectral gap. The introduced method is a one-step extrapolation technique that uses a linear combination of current and previous update steps to form a better approximation of the dominant eigenvector. The provided analysis shows the method converges exponentially with respect to the ratio between the two largest eigenvalues, which is also approximated during the process. An augmented technique is also introduced and is shown to stabilize the early stages of the iteration. Numerical examples are provided to illustrate the theory and demonstrate the methods.