Power Iteration Method for the Several Largest Eigenvalues and Eigenfunctions

A method for simultaneously obtaining the two largest eigenvalues and their associated eigenfunctions is demonstrated mathematically and empirically. The method uses estimates of the eigenvalue in two different regions rather than the single estimate traditionally used. The method can be generalized to obtain the several largest eigenfunctions, if those are desired as well. Additionally, it is shown that using multiple estimates of the eigenvalues accelerates the convergence of the eigenfunctions.