  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 exponential functions 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.
