Velocity, attenuation, and the quality factor of waves propagating in homogeneous media of arbitrary anisotropy and attenuation strength are calculated in high-frequency asymptotics using a stationary slowness vector, the vector evaluated at the stationary point of the slowness surface. This vector is generally complex-valued and inhomogeneous, meaning that the real and imaginary parts of the vector have different directions. The slowness vector can be determined by solving three coupled polynomial equations of the sixth order or by a nonlinear inversion. The procedure is simplified if perturbation theory is applied. The elastic medium is viewed as a background medium, and the attenuation effects are incorporated as perturbations. In the first-order approximation, the phase and ray velocities and their directions remain unchanged, being the same as those in the background elastic medium. The perturbation of the slowness vector is calculated by solving a system of three linear equations. The phase attenuation and phase Q-factor are linear functions of the perturbation of the slowness vector. Calculating the ray attenuation and ray Q-factor is even simpler than calculating the phase quantities because they are expressed in terms of perturbations of the medium without the need to evaluate the perturbation of the slowness vector. Numerical modeling indicates that the perturbations are highly accurate; the errors are less than 0.3% for a medium with a Q-factor of 20 or higher. The accuracy can be enhanced further by a simple modification of the first-order perturbation formulas.