  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.
