This talk shows the extension of the doubling algorithm, which originally is used to solve the algebraic Riccati equation, to solve the discretized Bethe-Salpeter eigenvalue problem, which actually is of importance in quantum chemistry and other academic subjects. We show the equivalence between the Bethe-Salpeter eigenvalue problem and the eigenvalue problem of a symplectic pencil, and then employ the doubling

algorithm to solve the corresponding eigenvalue problem of the symplectic pencil,where two matrices are merely computed in each iteration.