  We consider the so-called Keplerian periodogram, in which the putative
detectable signal is modelled by a highly non-linear Keplerian radial velocity
function, appearing in Doppler exoplanetary surveys. We demonstrate that for
planets on high-eccentricity orbits the Keplerian periodogram is far more
efficient than the classic Lomb-Scargle periodogram and even the multiharmonic
periodograms, in which the periodic signal is approximated by a truncated
Fourier series.
  We provide new numerical algorithm for computation of the Keplerian
periodogram. This algorithm adaptively increases the parameteric resolution
where necessary, in order to uniformly cover all local optima of the Keplerian
fit. Thanks to this improvement, the algorithm provides more smooth and
reliable results with minimized computing demands.
  We also derive a fast analytic approximation to the false alarm probability
levels of the Keplerian periodogram. This approximation has the form $(P
z^{3/2} + Q z) W \exp(-z)$, where $z$ is the observed periodogram maximum, $W$
is proportional to the settled frequency range, and the coefficients $P$ and
$Q$ depend on the maximum eccentricity to scan.
