We present a density functional theory (DFT)-based ab initio method for calculating selected portions of the vibrational spectrum, circumventing the need to evaluate the full Hessian. Our method is based on a combination of variational density functional perturbation theory and the Lanczos diagonalization method. The method is best suited to evaluating the extremal portion of the vibrational spectrum, but it can be adapted to target preselected regions of the spectrum.