Enhancing Nonlinear Equation Solutions Through the Combination of Variant Newton’s and Halley’s Methods

This work presents a new iterative method for solving single-variable nonlinear equations. The method achieves ninth-order convergence with just three derivative evaluations per step, offering both accuracy and lower computational cost. Unlike slower bracketing methods, it builds on faster open methods, though these may sometimes fail to converge. By blending ideas from Newton's and Halley's methods, the new approach provides strong performance, as shown by a detailed convergence analysis and MATLAB tests. Compared to existing techniques, it finds solutions in fewer steps and less time, making it especially effective for difficult nonlinear problems