  To solve the Bethe-Salpeter eigenvalue problem with distinct sizes, two
efficient methods, called {\Gamma}QR algorithm and {\Gamma}-Lanczos algorithm,
are proposed in this paper. Both algorithms preserve the special structure of
the initial matrix $H=\begin{bmatrix}A & B-\overline{B} &
-\overline{A}\end{bmatrix}$, 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.
