  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.
