A simple extrapolation method for clustered eigenvalues

TL;DR

This paper presents a simple one-step extrapolation method to accelerate convergence of eigenvalue computations, especially effective for problems with small spectral gaps, supported by theoretical analysis and numerical experiments.

Contribution

It introduces a novel one-step extrapolation technique for eigenvalue problems, improving convergence speed and stability over traditional methods.

Findings

Accelerates convergence to dominant eigenvalues

Effective for problems with small spectral gaps

Stabilizes early iteration stages

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.