Structure-Preserving ΓQR and Γ-Lanczos Algorithms for Bethe-Salpeter Eigenvalue Problems

Abstract

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 $H=\left[\begin{array}{ccc}A& B-\overline{B}& -\overline{A}\end{array}\right]$, 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.

31pages, 4figures