Steepest descent methods combining complex contour deformation with numerical quadrature provide an efficient and accurate approach for the evaluation of highly oscillatory integrals. However, unless the phase function governing the oscillation is particularly simple, their application requires a significant amount of a priori analysis and expert user input, to determine the appropriate contour deformation, and to deal with the non-uniformity in the accuracy of standard quadrature techniques associated with the coalescence of stationary points (saddle points) with each other, or with the endpoints of the original integration contour. In this paper we present a novel algorithm for the numerical evaluation of oscillatory integrals with general polynomial phase functions, which automates the contour deformation process and avoids the difficulties typically encountered with coalescing stationary points and endpoints. The inputs to the algorithm are simply the phase and amplitude functions, the endpoints and orientation of the original integration contour, and a small number of numerical parameters. By a series of numerical experiments we demonstrate that the algorithm is accurate and efficient over a large range of frequencies, even for examples with a large number of coalescing stationary points and with endpoints at infinity. As a particular application, we use our algorithm to evaluate cuspoid canonical integrals from scattering theory. A Matlab implementation of the algorithm is made available and is called PathFinder.

翻译：将复路径变形与数值求积相结合的最速下降法，为高振荡积分的评估提供了一种高效且精确的方法。然而，除非控制振荡的相位函数特别简单，否则其应用需要大量先验分析和专家用户输入，以确定适当的路径变形，并处理与驻点（鞍点）相互合并或与原始积分路径端点合并相关的标准求积技术精度非均匀性问题。本文提出了一种新颖的算法，用于数值评估具有一般多项式相位函数的振荡积分，该算法自动完成路径变形过程，并避免了通常遇到的驻点和端点合并困难。算法的输入仅为相位函数和振幅函数、原始积分路径的端点和方向，以及少量数值参数。通过一系列数值实验，我们证明了该算法在广泛的频率范围内准确且高效，即使对于具有大量合并驻点和无穷远端点的示例也是如此。作为一项特殊应用，我们使用该算法评估了散射理论中的尖点典型积分。该算法的Matlab实现已公开，称为PathFinder。