On the application of GMRES to oscillatory singular integral equations

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.