  In this paper the Lagrangian density of a purely kinetic k--essence that
models the behavior of dark energy described by four parameterized equations of
state proposed by Cooray \& Huterer (Astrophys. J. {\bf 513} L95, 1999), Zhang
\& Wu (Mod. Phys. Lett. A {\bf 27} 1250030, 2012), Linder (Phys. Rev. Lett.
{\bf 90} 091301, 2003), Efstathiou (Mon. Not. Roy. Astron. Soc. {\bf 310} 842,
2000), and Feng \& Lu (J. Cosmol. Astropart. Phys. {\bf 1111} 034, 2011) has
been reconstructed. This reconstruction is performed using the method outlined
by de Putter \& Linder (Astropart.\ Phys.\ {\bf 28} 263, 2007), which makes it
possible to solve the equations that relate the Lagrangian density of the
k--essence with the given equation of state (EoS) numerically. Finally, we
discuss the observational constraints for the models based on 586 SNIa data
points from the LOSS data set compiled by Ganeshalingam et al. (Mon. Not Roy.
Astron. Soc. {\bf 433} 2240, 2013).
