Using a superoperator formulation of linearized time-dependent density-functional theory, the dynamical polarizability of a system of interacting electrons is represented by a matrix continued fraction whose coefficients can be obtained from the nonsymmetric block-Lanczos method. The resulting algorithm, which is particularly convenient when large basis sets are used, allows for the calculation of the full spectrum of a system with a computational workload only a few times larger than needed for static polarizabilities within time-independent density-functional perturbation theory. The method is demonstrated with calculation of the spectrum of benzene, and prospects for its application to the large-scale calculation of optical spectra are discussed.