Reliable Computation of the Zeros of Solutions of Second Order Linear ODEs Using a Fourth Order Method

Abstract

A fourth order fixed point method to compute the zeros of solutions of second order homogeneous linear ODEs is obtained from the approximate integration of the Riccati equation associated with the ODE. The method requires the evaluation of the logarithmic derivative of the function and also uses the coefficients of the ODE. An algorithm to compute with certainty all the zeros in an interval is given which provides a fast, reliable, and accurate method of computation. The method is illustrated by the computation of the zeros of Gauss hypergeometric functions (including Jacobi polynomials) and confluent hypergeometric functions (Laguerre polynomials, Hermite polynomials, and Bessel functions included) among others. The examples show that typically 4 or 5 iterations per root are enough to provide more than 100 digits of accuracy, without requiring a priori estimations of the roots.