An Efficient Derivative Free Iterative Method and Condition Number of Nonlinear Equations Arising from Real World Phenomena

Recent researches have shown a great interest in developing novel numerical methods for solving nonlinear equations of the form arising from real world phenomena. However, very little attention has been given to the study of condition numbers which is an important aspect in measuring the sensitivity of the problem in response to slight perturbations in the input data. In this article, we present an efficient free derivative iterative scheme constructed by refining Newton-Raphson method standard form in which the derivative term is approximated by using finite difference scheme; hence making it derivative free. We also conducted an in-depth analysis of the condition numbers to explore sensitivity and efficiency comparisons between the proposed algorithm and existing methods for the given problems. Our investigation focused on iteration numbers, residuals, and convergence under mild error tolerances. Based on five numerical case studies, results revealed that the proposed Algorithm 2 outperforms the existing Algorithm 1 in terms of accuracy of the approximate solution. The results for the condition numbers indicated that all problems considered were ill-conditioned, highlighting the significance of studying condition numbers in the context of solving nonlinear equations.