Finite Difference Weerakoon-Fernando Method to solve nonlinear equations without using derivatives

TL;DR

This paper introduces the Finite Difference Weerakoon-Fernando Method (FDWFM), an efficient derivative-free algorithm for solving nonlinear equations with faster convergence than existing methods like secant.

Contribution

The paper formulates a new derivative-free root-finding method based on the Weerakoon-Fernando approach, achieving near quadratic convergence and applicability to complex roots and systems.

Findings

FDWFM converges faster than secant method without derivatives.

FDWFM achieves a convergence order close to 2.5.

Effective for complex roots and systems of nonlinear equations.

Abstract

This research was mainly conducted to explore the possibility of formulating an efficient algorithm to find roots of nonlinear equations without using the derivative of the function. The Weerakoon-Fernando method had been taken as the base in this project to find a new method without the derivative since Weerakoon-Fernando method gives 3rd order convergence. After several unsuccessful attempts we were able to formulate the Finite Difference Weerakoon-Fernando Method (FDWFM) presented here. We noticed that the FDWFM approaches the root faster than any other existing method in the absence of the derivatives as an example, the popular nonlinear equation solver such as secant method (order of convergence is 1.618) in the absence of the derivative. And the FDWFM had three function evaluations and secant method had two function evaluations. By implementing FDWFM on nonlinear equations with complex roots and also on systems of nonlinear equations, we received very encouraging results. When applying the FDWFM to systems of nonlinear equations, we resolved the involvement of the Jacobian problem by following the procedure in the Broyden's method. The computational order of convergence of the FDWFM was close to 2.5 for all these cases. This will undoubtedly provide scientists the efficient numerical algorithm, that doesn't need the derivative of the function to solve nonlinear equations, that they were searching for over centuries.