On the numerical approximation for Fourier-type highly oscillatory integrals with Gauss-type quadrature rules
In this paper, we present an improved numerical steepest descent method for the approximation of Fourier-type highly oscillatory integrals. Based on the previous numerical steepest descent method, the new method used the integrand information at endpoints and stationary points. The asymptotic order is given that is improved both for the case of stationary points and stationary points free. Several numerical examples are presented which show the high efficiency of the proposed method. Numerical results support our theoretical analyses.