Computational performance of classical and hybrid root-finding methods for real-world nonlinear models

Abstract We offer and discuss a two-point root-finding technique which is a hybrid approach that combines Newton’s method with the Secant method. The hybrid version can be represented as $$\begin{aligned} x_{n +1} = x_n -\frac{2 f(x_n)}{\,f'(x_n) + \dfrac{f(x_n)-f(x_{n-1})}{(x_n -x_{n-1})}}. \end{aligned}$$ This formulation uses derivative’s information along with a finite-difference approximation to the derivatives. We give a local convergence study using standard hypotheses and compare the practical performance of the hybrid method to that of Newton, Secant and the Weerakoon–Fernando method (WFM) on a set of real-world nonlinear equations in engineering and physical applications. Numerical experiments give counts of iterations, counts of evaluations of functions and derivatives, and CPU times along with sensitivity to initial guesses. Results show that the proposed hybrid scheme achieves a desirable balance between convergence rate and robustness in the chosen models, and therefore, is especially applicable in the computational context of the contemporary world, where predictable convergence and stability are vital.