  The treatment of the Random-Phase Approximation Hamiltonians, encountered in
different frameworks, like Time-Dependent Density Functional Theory or
Bethe-Salpeter equation, is complicated by their non-Hermicity. Compared to
their Hermitian Hamiltonian counterparts, computational methods for the
treatment of non-Hermitian Hamiltonians are often less efficient and less
stable, sometimes leading to the breakdown of the method. Recently [Gr\"uning
et al. Nano Lett. {\bf 8}, 2820 (2009)], we have identified that such
Hamiltonians are usually pseudo-Hermitian. Exploiting this property, we have
implemented an algorithm of the Lanczos type for random-Phase Approximation
Hamiltonians that benefits from the same stability and computational load as
its Hermitian counterpart, and applied it to the study of the optical response
of carbon nanotubes. We present here the related theoretical grounds and
technical details, and study the performance of the algorithm for the
calculation of the optical absorption of a molecule within the Bethe-Salpeter
equation framework.
