Some Iterative Methods For Solving System Of Nonlinear Equations

As is known to all, it is of great significance to study how to solve a system of nonlinear equations since a lot of practical problems in engineering, economics, information security, and dynamics are eventually transformed into the problems of finding the solutions of the systems of nonlinear equations. This thesis constructs several iterative schemes to solve the system of nonlinear equations. The main contributions include:Firstly, we present three new iterative methods for solving the system of nonlinear equations based on a modified two-step Newton’s method and different Newton-Cotes quadrature formulaes. We prove that these new methods are of the convergence of fifth order. Some numerical examples are given to show that the new methods outperform the existing methods.Secondly, a general scheme for solving the system of nonlinear equations is developed with the parameters α and β which can increase the order of convergence of any given iterative methods. If α+β=1with β≠0, the proposed approach is of convergence order p+2. Moreover, it is pointed out that the iterative methods presented in [31] and [38] are corresponding to the cases α=0,β=1and α=1,β=0, respectively. We get a sixth order method by applying the general scheme (taking α=1/2,β=1/2) to a fourth order method. Finally, some numerical examples are given to compare with different schemes.Thirdly, the modified Newton’s method and the pseudocomposition method presented by Cordero are combined to design a new iterative scheme of the convergence order at least3p for solving the system of nonlinear equations. The efficiency index of the new method is compared with those of the existing methods. Finally, some numerical examples are given to illustrate the effectiveness of the new method.