Structure-preserving ΓQR and Γ-Lanczos algorithms for Bethe-Salpeter eigenvalue problems

To solve the Bethe-Salpeter eigenvalue problem with distinct sizes, two efficient methods, called ΓQR algorithm and Γ-Lanczos algorithm, are proposed in this paper. Both algorithms preserve the special structure of the initial matrix ℋ=AB−B¯−A¯, resulting the computed eigenvalues and the associated eigenvectors still hold the properties similar to those of ℋ. Theorems are given to demonstrate the validity of the proposed two algorithms in theory. Numerical results are presented to illustrate the superiorities of our methods.