Modified Halley's method for computation of zeros of solution of second order ODEs

TL;DR

This paper introduces a modified third-order Halley's iterative method for efficiently finding all zeros of solutions to second order ODEs, with proven convergence and applications to quadrature node computation.

Contribution

A novel modified Halley's method with fixed function for second order ODE zeros, maintaining third order convergence and improved efficiency.

Findings

The modified method retains third order convergence.

Algorithms for quadrature nodes and weights are developed.

Numerical results show superior efficiency over recent methods.

Abstract

This paper develops an efficient iterative method for computing all zeros of solutions of second order ordinary differential equations. A third order Halleys method is first derived by approximating the solution of an associated Riccati differential equation. To improve computational efficiency, a modified Halleys method is proposed by fixing one of the functions in Halleys scheme as a constant. The modified Halleys method also retains third order convergence. Based on the behavior of the coefficients of the second order ODE, nonlocal convergence results are established for both Halleys and modified Halleys methods. Suitable initial guesses for computing all zeros of solutions of second order ODEs in a given interval are also presented for both methods. Furthermore, algorithms based on the modified Halleys method are developed for to compute all nodes and weights for Gauss Legendre and Gauss Hermite quadratures. A comparative numerical study with recent methods demonstrates the efficiency of the proposed algorithms.