  The Bethe-Salpeter equation (BSE) that results from the GW approximation to
the self-energy is a frequency-dependent (nonlinear) eigenvalue problem due to
the dynamically screened Coulomb interaction between electrons and holes. The
computational time required for a numerically exact treatment of this frequency
dependence is $O(N^6)$, where $N$ is the system size. To avoid the common
static screening approximation, we show that the full-frequency dynamical BSE
can be exactly reformulated as a frequency-independent eigenvalue problem in an
expanded space of single and double excitations. When combined with an
iterative eigensolver and the density fitting approximation to the electron
repulsion integrals, this reformulation yields a dynamical BSE algorithm whose
computational time is $O(N^5)$, which we verify numerically. Furthermore, the
reformulation provides direct access to excited states with dominant double
excitation character, which are completely absent in the spectrum of the
statically screened BSE. We study the $2^1A_\mathrm{g}$ state of butadiene,
hexatriene, and octatetraene and find that GW/BSE overestimates the excitation
energy by about 1.5-2 eV and significantly underestimates the double excitation
character.
