Third-order Halley-type iterative method with positive and bounded correction function

Abstract

In this paper, we propose a new one-point iterative method for finding simple roots of nonlinear equations by modifying Halley's method. Its correction function is positive and bounded on $\mathbb{R}$ and matches the local expansion of Halley's correction function near a simple root. The method avoids the singularity and sign reversal of Halley's correction function while retaining at least third-order local convergence. We also establish sufficient conditions for global convergence. Numerical experiments show that the proposed method achieves high convergence success rates and favorable iteration counts over a wide range of initial guesses. Further experiments near the singularity of Halley's correction function demonstrate robust convergence behavior of the proposed method. An application to the van der Waals equation illustrates the effectiveness of the proposed method.