  We consider the use of rational basis functions to compute the scattering and
inverse scattering transforms associated with the AKNS system. The proposed
numerical forward scattering transform computes the solution of the AKNS system
that valid on the entire real axis and thereby computes a reflection
coefficient at a point by solving a single linear system. The proposed
numerical inverse scattering transform makes use of a novel improvement in the
rational function approach to the oscillatory Cauchy operator, enabling the
efficient solution of certain Riemann--Hilbert problems without contour
deformations. The later development enables access to high-precision
computations and this is demonstrated on the inverse scattering transform for
the one-dimensional Schr\"odinger operator with a $\mathrm{sech}^2$ potential.
