A new iteration method with cubic convergence to solve nonlinear algebraic equations

In this paper, a new iteration scheme is proposed to solve the roots of an algebraic equation f(x) = 0. Given an initial guess, x0, the roots of the equation can be obtained using the following iteration scheme:xn+1=xn+-f′(xn)±f′2(xn)-2f(xn)f″(xn)f″(xn).This iteration scheme has unique convergence characteristics different from the well-known Newton’s method. It is shown that this iteration method has cubic local convergence in the neighborhood of the root. Using this scheme, real or complex roots for specific algebraic equations can be found. Because there are two iteration directions, for a given initial guess, two solutions can be found for certain algebraic equations with multiple roots. Examples are presented and compared with other methods.