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：结合复路径形变与数值求积的最速下降法为高振荡积分评估提供了高效且精确的途径。然而，除非控制振荡的相位函数具有极简形式，否则其应用需大量先验分析与专家人工干预：既要确定合适的路径形变方案，又要处理标准求积技术中因驻点（鞍点）相互合并或与初始积分路径端点合并导致的精度非均匀性问题。本文提出一种针对一般多项式相位函数振荡积分的新型数值算法，该算法自动实现路径形变过程，并规避了驻点与端点合并时常见的困难。算法输入仅需相位函数与振幅函数、原始积分路径端点与方向，以及少量数值参数。通过系列数值实验证明，该算法在大频率范围内保持高精度与高效率，即使面对大量驻点合并及无穷远端点情形仍表现稳定。作为特定应用，我们将算法用于评估散射理论中的尖点典型积分。算法已通过Matlab实现并以PathFinder命名公开。