A novel cubically convergent iterative method for computing complex roots of nonlinear equations

摘要

A fast and simple iterative method with cubic convergent is proposed for the determination of the real and complex roots of any function F(x) = 0. The idea is based upon passing a defined function G(x) tangent to F(x) at an arbitrary starting point. Choosing G(x) in the form of xk or kx, where k is obtained for the best correlation with the function F(x), gives an added freedom, which in contrast to all existing methods, accelerates the convergence. Also, this new method can find complex roots just by a real initial guess. This is in contrast to many other methods like the famous Newton method that needs complex initial guesses for finding complex roots. The proposed method is compared to some new and famous methods like Newton method and a modern solver that is fsolve command in MATLAB. The results show the effectiveness and robustness of this new method as compared to other methods.