The 2n+1 theorem and the density-functional perturbation theory have been used to calculate anharmonic force constants completely ab initio. Explicit expressions for the anharmonic coupling constants are presented, i.e., for the third-order derivatives of the total energy with respect to atomic displacements. Using the harmonic as well as the anharmonic results the phonon linewidth of Ge and Si as a function of temperature has been calculated for various branches and various (in particular, for nonvanishing) wave vectors completely ab initio.