Computing the real zeros of cylinder functions and the roots of the equation xC '(nu)(x) + gamma C-v(x)=0
Fast methods to compute the zeros of general cylinder functions C-nu(x) = cos alpha J(nu)(x) - sin alpha Y-nu(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 omega(x) = nu'u(x) = x(gamma)C(nu)(x). we also construct a globally convergent fourth order method for the evaluation of the roots of xC'(nu)(x) + gamma C-nu(x) = 0, and in particular for the first derivative of cylinder functions. The method holds for any real values of nu, alpha and gamma 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 C-nu'(x) and the function xC(nu)'(x) + gamma C-nu(x), a double root or a nearly degenerate pair of real roots may exist for some parameter values; the method computes reliably such roots. (C) 2012 Elsevier Ltd. All rights reserved.
second order odes bessel functions nonlinear equations fixed point method sturm theorems