Multiple Extremal Eigenpairs by the Power Method

TL;DR

This paper introduces refined power method techniques for efficiently computing multiple extremal eigenpairs of large matrices, demonstrated on models relevant to physics and nuclear engineering.

Contribution

The paper presents novel refinements to the power method enabling calculation of multiple extremal eigenpairs, including up to the 10th, using an observation by Booth.

Findings

Successfully computed the two extremal eigenpairs of various large matrices

Demonstrated effectiveness on matrices from physics models

Achieved up to the 10th eigenpair calculation for simple test problems

Abstract

We report the production and benchmarking of several refinements of the power method that enable the computation of multiple extremal eigenpairs of very large matrices. In these refinements we used an observation by Booth that has made possible the calculation of up to the 10$^{th}$ eigenpair for simple test problems simulating the transport of neutrons in the steady state of a nuclear reactor. Here, we summarize our techniques and efforts to-date on determining mainly just the two largest or two smallest eigenpairs. To illustrate the effectiveness of the techniques, we determined the two extremal eigenpairs of a cyclic matrix, the transfer matrix of the two-dimensional Ising model, and the Hamiltonian matrix of the one-dimensional Hubbard model.