Computing the real zeros of cylinder functions and the roots of the equation [formula omitted

Abstract

Fast methods to compute the zeros of general cylinder functions Cν(x)=cosαJν(x)−sinαYν(x) in real intervals can be obtained from an approximate integration of the second order ODE satisfied by these functions, leading to fourth order methods with global convergence. By considering the second order ODE satisfied by the function w(x)=v′(x), v(x)=xγCν(x), we also construct a globally convergent fourth order method for the evaluation of the roots of xCν′(x)+γCν(x)=0, and in particular for the first derivative of cylinder functions. The method holds for any real values of ν, α and γ and it does not require a priori estimations for initiating the iterations. Fifteen digit accuracy is generally reached with only 1–3 iterations per each simple root. For the derivative Cν′(x) and the function xCν′(x)+γCν(x), a double root or a nearly degenerate pair of real roots may exist for some parameter values; the method computes reliably such roots.