Efficient third-order iterative algorithms for computing zeros of special functions

TL;DR

This paper introduces a new third-order iterative algorithm for accurately finding zeros of solutions to second-order differential equations, with proven convergence and demonstrated effectiveness through numerical simulations.

Contribution

It develops a novel third-order method based on Riccati equation approximation, ensuring convergence and applicability to various special functions.

Findings

Method converges reliably for Legendre and Hermite polynomials.

Numerical results confirm the efficiency and accuracy of the proposed algorithm.

Comparative analysis shows improvements over recent methods.

Abstract

This manuscript presents a novel and reliable third-order iterative procedure for computing the zeros of solutions to second-order ordinary differential equations. By approximating the solution of the related Riccati differential equation using the trapezoidal rule, this study has derived the proposed third-order method. This work establishes sufficient conditions to ensure the theoretical non-local convergence of the proposed method. This study provides suitable initial guesses for the proposed third-order iterative procedure to compute all zeros in a given interval of the solutions to second-order ordinary differential equations. The orthogonal polynomials like Legendre and Hermite, as well as the special functions like Bessel, Coulomb wave, confluent hypergeometric, and cylinder functions, satisfy the proposed conditions for convergence. Numerical simulations demonstrate the effectiveness of the proposed theory. This work also presents a comparative analysis with recent studies.