Preconditioned GMRES for oscillatory integrals

Abstract

None of the existing methods for computing the oscillatory integral ∫ab f(x)eiωg(x) dx achieve all of the following properties: high asymptotic order, stability, avoiding the computation of the path of steepest descent and insensitivity to oscillations in f. We present a new method that satisfies these properties, based on applying the GMRES algorithm to a preconditioned differential operator.