We describe a method for the numerical evaluation of the angular prolate spheroidal wave functions of the first kind of order zero. It is based on the observation that underlies the WKB method, namely that many second order differential equations admit solutions whose logarithms can be represented much more efficiently than the solutions themselves. However, rather than exploiting this fact to construct asymptotic expansions of the prolate spheroidal wave functions, our algorithm operates by numerically solving the Riccati equation satisfied by their logarithms. Its running time grows much more slowly with bandlimit and characteristic exponent than standard algorithms. We illustrate this and other properties of our algorithm with numerical experiments.

我们描述了一种数值评估第一类零阶角长球面波函数的方法。它基于作为 WKB 方法基础的观察，即许多二阶微分方程允许解，其对数可以比解本身更有效地表示。然而，我们的算法不是利用这一事实来构造长球面波函数的渐近展开，而是通过数值求解由它们的对数满足的 Riccati 方程来运行。它的运行时间随着带宽限制和特征指数的增长比标准算法慢得多。我们通过数值实验来说明我们算法的这个和其他特性。