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, providing up to a $\sim$50 times improvement in
computational performance 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 for further accelerating the PTA signal
evaluations for narrow-band PTA datasets. These results are crucial for
searching for eccentric SMBHBs in large PTA datasets. We have implemented these
results in the GWecc package and can be readily accessed from the popular
ENTERPRISE package to search for such signals in PTA datasets.
