Some numerical methods for solving nonlinear equations by using decomposition technique

Keywords: Decomposition technique Iterative method Convergence Newton method Auxiliary function Coupled system of equations a b s t r a c t In this paper, we use the system of coupled equations involving an auxiliary function together with decomposition technique to suggest and analyze some new classes of iterative methods for solving nonlinear equations. These new methods include the Halley method and its variant forms as special cases. Various numerical examples are given to illustrate the efficiency and performance of the new methods. These new iterative methods may be viewed as an addition and generalization of the existing methods for solving nonlinear equations. It is well known that a wide class of problems, which arises in diverse disciplines of mathematical and engineering science can be studied by the nonlinear equation of the form f ðxÞ ¼ 0. Numerical methods for finding the approximate solutions of the nonlinear equation are being developed by using several different techniques including Taylor series, quadrature formulas, homotopy and decomposition techniques, see [1–17] and the references therein. In this paper, we use alternative decomposition technique to suggest the main iterative schemes which generates the iterative methods of higher order. First of all, we rewrite the given nonlinear equation along with the auxiliary function as an equivalent coupled system of equations using the Taylor series. This approach enables us to express the given nonlinear equation as sum of linear and nonlinear equations. This way of writing the given equation is known as the decomposition and plays the central role in suggesting the iterative methods for solving nonlinear equations f ðxÞ ¼ 0. In this work, we use the system of coupled equations to express the given nonlinear equations as a sum of linear and non-linear operators involving the auxiliary function gðxÞ. This auxiliary function helps to deduce several iterative methods for solving nonlinear equations. The effectiveness and efficiency of the auxiliary function can be observed in the next section for deriving the robust iterative methods for solving nonlinear equations. Results obtained in this paper, suggest that this new technique of decomposition is a promising tool. In section 2, we sketch out the main ideas of this technique and suggest some multi-step iterative methods for solving nonlinear equations. One can notice that if the derivative of the function vanishes, that is f 0 ðx n Þ ¼ 0, during the iterative process, then the sequence generated by the …