Two-point iterative methods for solving nonlinear equations

In this study, a new root-finding technique for solving nonlinear equations is proposed. Then, two new more algorithms are derived from this new technique by employing the Adomian decomposition method (ADM). These three algorithms require two starting values that do not necessarily bracketing the root of a given nonlinear equation, however, when the starting values are closed enough or bracketed the root, then the proposed methods converge to the root faster than the secant method. Another advantage over all iterative methods is that; the proposed methods usually converge to two distinct roots when the handled nonlinear equation has more than one root, that is, the odd iterations of the new techniques converge to a root and the even iterations converge to another root. Some numerical examples, including a sine-polynomial equation, are solved by using the proposed methods and compared with the secant method; perfect agreements were found.