Numerical Comparison of the Mamadu Δ³ Iterative Scheme and Newton–raphson Method for Nonlinear Equations

Abstract

This study numerically compares the Mamadu Δ³ iterative scheme with the Newton–Raphson method for solving nonlinear polynomial equations. The comparison considers their mathematical formulation, iterative implementation, convergence behaviour, derivative requirements, and numerical performance. Two nonlinear polynomial equations are used as test problems, with initial approximations selected within intervals containing the required positive roots. Both methods are applied iteratively until stable approximations are obtained to the prescribed numerical accuracy. For the problems considered, both methods converge to consistent positive-root approximations. The Mamadu Δ³ scheme requires fewer iterations, consistent with its stated fourth-order convergence, whereas the Newton–Raphson method uses only first-derivative information and has a simpler computational structure. The numerical comparison therefore distinguishes iteration efficiency from per-iteration complexity: the Mamadu Δ³ scheme reduces the number of iterations for the selected examples, while Newton–Raphson retains an implementation advantage because of its lower derivative requirement. The results also indicate that the choice of the initial approximation influences the practical behaviour of both iterative procedures. Overall, the study provides a direct numerical comparison of the two methods under the selected test conditions and highlights the trade-off between convergence order, derivative evaluation, and implementation simplicity when solving nonlinear equations. The findings are therefore interpreted within the scope of the two reported polynomial examples.