Derivative-Free Finite-Difference Homeier Method for Nonlinear Models

Abstract

An efficient derivative-free method for determining roots with respect to nonlinear equations was implemented in this paper. The third-order Homeier’s method has been taken as the basis for this work, which can be derived by employing Newton’s theorem for the inverse function as well as deriving a new class of cubically convergent Newton-type methods. Several nonlinear problems, including nonlinear equations, complex equations, and nonlinear systems of equations, have been considered in order to perform a comparison with regard to the efficiency of the suggested method to other popular derivative-free schemes. Results show that the proposed method Derivative-Free Homeier method (DFH) outperformed the considered published methods. The DFH needs fewer iterations to achieve the desired solution, with an order of convergence of about 2.4, which is higher than the convergence order with regard to the methods that were compared. Here, one of the popular nonlinear equation solvers used to compare with our proposed method is the secant method having a convergence order of 1.618 in the derivative’s absence. Furthermore, by adhering to the steps of Broyden’s method when utilizing the DFH to solve systems of nonlinear equations, the Jacobian problem can be averted. Therefore, the DFH can be considered as an uppermost method giving faster convergence to determine the nonlinear equations’ roots with no derivative for uni-variate nonlinear equations having complex roots, including multivariate systems of nonlinear equations.