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Quantifying materials' dynamical responses to external electromagnetic fieldsis central to understanding their physical properties. Here we present animplementation of the density functional perturbation theory for thecomputation of linear susceptibilities using the projector augmented-wavemethod. The Sternheimer equations are solved self-consistently through a nestediterative procedure to compute the first-order wavefunctions, from which thelinear susceptibilities are obtained. As a demonstration, we compute the spinwave spectral functions of two magnetic metals. The computed magnon spectra forhalf-metallic CrO$_2$ and a Heusler intermetallic Cu$_2$MnAl show gaplessGoldstone modes when spin rotation symmetry is preserved and display reasonableagreement with available experimental data. The Landau damping is computed tobe small in CrO$_2$, but significant in Cu$_2$MnAl producing an asymmetricLorentzian spectral lineshape. The access to linear susceptibilities as well asfirst-order wavefunctions offers a range of novel possibilities in quantitativeunderstanding of materials' electronic properties from \textit{ab initio}methods.

An implementation of the density functional perturbation theory in the PAW framework