The Bethe-Salpeter eigenvalue problem is a dense structured eigenvalueproblem arising from discretized Bethe-Salpeter equation in the context ofcomputing exciton energies and states. A computational challenge is that atleast half of the eigenvalues and the associated eigenvectors are desired inpractice. We establish the equivalence between Bethe-Salpeter eigenvalueproblems and real Hamiltonian eigenvalue problems. Based on theoreticalanalysis, structure preserving algorithms for a class of Bethe-Salpetereigenvalue problems are proposed. We also show that for this class of problemsall eigenvalues obtained from the Tamm-Dancoff approximation are overestimated.In order to solve large scale problems of practical interest, we discussparallel implementations of our algorithms targeting distributed memorysystems. Several numerical examples are presented to demonstrate the efficiencyand accuracy of our algorithms.
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