Phase function methods for second order linear ordinary differential equations with turning points

It is well known that second order linear ordinary differential equations with slowly varying coefficients admit slowly varying phase functions. This observation is the basis of the Liouville-Green method and many other techniques for the asymptotic approximation of the solutions of such equations. More recently, it was exploited by the author to develop a highly efficient solver for second order linear ordinary differential equations whose solutions are oscillatory. In many cases of interest, that algorithm achieves near optimal accuracy in time independent of the frequency of oscillation of the solutions. Here we show that, after minor modifications, it also allows for the efficient solution of second order differential equation equations which have turning points. That is, it is effective in the case of equations whose solutions are oscillatory in some regions and behave like linear combinations of increasing and decreasing exponentials in others. We present the results of numerical experiments demonstrating the properties of our method, including some which show that it can used to evaluate many classical special functions in time independent of the parameters on which they depend.