  Pulsar Timing Array (PTA) experiments are expected to be sensitive to
gravitational waves (GWs) emitted by individual supermassive black hole
binaries (SMBHBs) inspiralling along eccentric orbits. We compare the
computational cost of different methods of computing the PTA signals induced by
relativistic eccentric SMBHBs, namely approximate analytic expressions, Fourier
series expansion, post-circular expansion, and numerical integration. We show
that the fastest method for evaluating PTA signals is by using the approximate
analytic expressions, which provides up to ~50 times improvement in
computational speed over the alternative methods. We investigate the accuracy
of the approximate analytic expressions by employing a mismatch metric valid
for PTA signals. We show that this method is accurate within the region of the
binary parameter space that is of interest to PTA experiments. We introduce a
spline-based method to further accelerate the PTA signal evaluations for
narrowband PTA datasets. The efficient methods for computing the eccentric
SMBHB-induced PTA signals were implemented in the GWecc.jl package and can be
readily accessed from the popular ENTERPRISE package to search for such signals
in PTA datasets. Further, we simplify the eccentric SMBHB PTA signal expression
for the case of a single-pulsar search and demonstrate our computationally
efficient methods by performing a single-pulsar search in the 12.5-year
NANOGrav narrowband dataset of PSR J1909-3744 using the simplified expression.
These results will be crucial for searching for eccentric SMBHBs in large PTA
datasets.
