Modified Halley's method for computation of zeros of solution of second order ODEs
This work addresses a specific computational bottleneck in numerical analysis for researchers and practitioners dealing with ODEs and quadrature methods, representing an incremental improvement over existing methods.
The paper tackles the problem of computing zeros of solutions for second-order ordinary differential equations by proposing a modified Halley's method that improves computational efficiency while maintaining third-order convergence, and demonstrates its effectiveness through comparative numerical studies for applications like Gauss quadratures.
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.