The complexity of the solutions of a differential equation of the form $y''(t) + \omega^2 q(t) y(t) =0$ depends not only on that of the coefficient $q$, but also on the magnitude of the parameter $\omega$. In the most widely-studied case, when $q$ is positive, the solutions of equations of this type oscillate at a frequency that increases linearly with $\omega$ and standard ODE solvers require $\mathcal{O}\left(\omega\right)$ time to calculate them. It is well known, though, that phase function methods can be used to solve such equations numerically in time independent of $\omega$. Unfortunately, the running time of these methods increases with $\omega$ when they are applied in the commonly-occurring case in when $q$ has zeros in the solution domain (i.e., when the differential equation has turning points). Here, we introduce a generalized phase function method adapted to equations with simple turning points. More explicitly, we show the existence of slowly-varying ``Airy phase functions'' which represent the solutions of such equations at a cost which is independent of $\omega$ and describe a numerical method for calculating these Airy phase functions in time independent of $\omega$. Using our method, initial or boundary value problems for a large class of second order linear ordinary differential equations with turning points whose coefficients depend on a parameter $\omega$ can be solved in time independent of $\omega$. We also give the results of numerical experiments conducted to demonstrate the properties of our method, including one in which we used our algorithm to rapidly calculate associated Legendre functions of a wide range of orders and degrees.

翻译：形如 $y''(t) + \omega^2 q(t) y(t) =0$ 的微分方程解的复杂度不仅取决于系数 $q$ 的复杂性，也依赖于参数 $\omega$ 的大小。在最广泛研究的情形中，当 $q$ 为正时，此类方程的解以随 $\omega$ 线性增长的频率振荡，标准的常微分方程求解器需要 $\mathcal{O}\left(\omega\right)$ 的时间来计算它们。然而众所周知，相位函数方法可用于在时间上与 $\omega$ 无关的情况下数值求解此类方程。不幸的是，当这些方法应用于 $q$ 在求解域内存在零点（即微分方程具有转向点）这一常见情形时，其运行时间会随 $\omega$ 增加。本文提出了一种适用于具有简单转向点方程的广义相位函数方法。更具体地说，我们证明了存在缓慢变化的“艾里相位函数”，它们能以与 $\omega$ 无关的代价表示此类方程的解，并描述了一种在时间上与 $\omega$ 无关的情况下计算这些艾里相位函数的数值方法。利用我们的方法，对于一大类系数依赖于参数 $\omega$ 且具有转向点的二阶线性常微分方程，其初值或边值问题可以在时间上与 $\omega$ 无关的情况下求解。我们还给出了数值实验的结果，以展示我们方法的特性，其中包括使用我们的算法快速计算了大范围阶数与次数的关联勒让德函数。