  We report total energy calculations for different crystal structures of
NaNbO$_3$ over a range of unit cell volumes using the all-electron
full-potential (L)APW method. We employed both the local-density approximation
(LDA) and the Wu-Cohen form of the generalized gradient approximation (GGA-WC)
to test the accuracy of these functionals for the description of the complex
structural behavior of NaNbO$_3$. We found that LDA not only underestimates the
equilibrium volume of the system but also predicts an incorrect ground state
for this oxide. The GGA-WC functional, on the other hand, significantly
improves the equilibrium volume and provides relative phase stability in better
agreement with experiments. We then use the GGA-WC functional for the
calculation of the phonon dispersion curves of cubic NaNbO$_3$ to identify the
presence of structural instabilities in the whole Brillouin zone. Finally, we
report comparative calculations of structural instabilities as a function of
volume in NaNbO$_3$ and KNbO$_3$ to provide insights for the understanding of
the structural behavior of K$_{1-x}$Na$_x$NbO$_3$ solid solutions.
