Derivative-Free Method for Nonlinear Systems with Some Real-life Applications

This study implements a very effective derivative-free approach for root determination in the context of nonlinear equations. The use of Homeier’s third-order technique, which is based on the application of Newton’s theorem to the inverse function, as well as the development of a new category of Newton-type procedures that exhibit cubic convergence, is the basis of this study. A variety of non-linear problems, such as non-linear equations and non-linear systems of equations, have been examined to examine the efficacy of the proposed method in contrast to other widely used derivative-free approaches. The results suggest that the proposed method outperforms the currently examined techniques published. The scheme given in this study demonstrates a reduced number of iterations acquired in attaining the intended result, exhibiting a convergence order of around 2.5. This convergence order surpasses the convergence order observed in comparable approaches. Moreover, by following the prescribed procedure of the Broyden technique while implementing the recommended approach in the solution of nonlinear equation systems, the problem pertaining to the Jacobian can be circumvented. Hence, the technique suggested may be regarded as a prominent approach that achieves rapid convergence in determining the roots of non-linear equations without the use of derivatives. In practice, we employ the novel approach to address practical challenges such as the mixed Hammerstein integral equation, the discretized Poisson equation, and the Weierstrass method. Through comparisons and illustrative examples, it is evident that the proposed method proves to be effective and on par with existing techniques of equivalent order.