  We present a new structure preserving Lanczos algorithm for approximating the
optical absorption spectrum in the context of solving full Bethe--Salpeter
equation without Tamm--Dancoff approximation. The new algorithm is based on a
structure preserving Lanczos procedure, which exploits the special block
structure of Bethe--Salpeter Hamiltonian matrices. A recently developed
technique of generalized averaged Gauss quadrature is incorporated to
accelerate the convergence. We also establish the connection between our
structure preserving Lanczos procedure with several existing Lanczos procedures
developed in different contexts. Numerical examples are presented to
demonstrate the effectiveness of our Lanczos algorithm.
