Shifted GMRES for oscillatory integrals

Abstract

None of the existing methods for computing the oscillatory integral $${\int_a^b f(x){\rm e}^{{\rm i}\omega g(x)}{\rm d}x}$$achieve all of the following properties: high asymptotic order, stability, avoiding deformation into the complex plane and insensitivity to oscillations in f. We present a new method that satisfies these properties, based on applying the gmres algorithm to a shifted linear differential operator.