  We present a new method for the numerical solution of singular integral
equations on the real axis. The method's value stems from an explicit formula
for the Cauchy integral of a complex exponential multiplied by a rational
function. Additionally, the inner product of such functions is computed
explicitly. With these tools, the GMRES algorithm is applied to both
non-oscillatory and oscillatory singular integral equations. Ideas from
Fredholm theory and Riemann--Hilbert problems are used to motivate
preconditioners for these singular integral equations. A dramatic acceleration
in convergence is realized. This presents a strong link between the theory of
singular integral equations and the numerical analysis of such equations.
Furthermore, this method presents a first step towards a solver for the inverse
scattering transform that does not require the deformation of a
Riemann--Hilbert problem.
