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We report the production and benchmarking of several refinements of the powermethod that enable the computation of multiple extremal eigenpairs of verylarge matrices. In these refinements we used an observation by Booth that hasmade possible the calculation of up to the 10$^{th}$ eigenpair for simple testproblems simulating the transport of neutrons in the steady state of a nuclearreactor. Here, we summarize our techniques and efforts to-date on determiningmainly just the two largest or two smallest eigenpairs. To illustrate theeffectiveness of the techniques, we determined the two extremal eigenpairs of acyclic matrix, the transfer matrix of the two-dimensional Ising model, and theHamiltonian matrix of the one-dimensional Hubbard model.

Multiple Extremal Eigenpairs by the Power Method