An Iterative Scheme of Arbitrary Odd Order and Its Basins of Attraction for Nonlinear Systems

In this paper,we propose a fifth-order scheme for solving systems of nonlinear equations.The convergence analysis of the proposed technique is discussed.The proposed method is generalized and extended to be of any odd order of the form 2n1.The scheme is composed of three steps,of which the first two steps are based on the two-step Homeier’s method with cubic convergence,and the last is a Newton step with an appropriate approximation for the derivative.Every iteration of the presented method requires the evaluation of two functions,two Fréchet derivatives,and three matrix inversions.A comparison between the efficiency index and the computational efficiency index of the presented scheme with existing methods is performed.The basins of attraction of the proposed scheme illustrated and compared to other schemes of the same order.Different test problems including large systems of equations are considered to compare the performance of the proposed method according to other methods of the same order.As an application,we apply the new scheme to some real-life problems,including the mixed Hammerstein integral equation and Burgers’equation.Comparisons and examples show that the presented method is efficient and comparable to the existing techniques of the same order.

Dynamical Comparison of Several Third-Order Iterative Methods for Nonlinear Equations

There are several ways that can be used to classify or compare iterative methods for nonlinear equations,for instance;order of convergence,informational efficiency,and efficiency index.In this work,we use another way,namely the basins of attraction of the method.The purpose of this study is to compare several iterative schemes for nonlinear equations.All the selected schemes are of the third-order of convergence and most of them have the same efficiency index.The comparison depends on the basins of attraction of the iterative techniques when applied on several polynomials of different degrees.As a comparison,we determine the CPU time(in seconds)needed by each scheme to obtain the basins of attraction,besides,we illustrate the area of convergence of these schemes by finding the number of convergent and divergent points in a selected range for all methods.Comparisons confirm the fact that basins of attraction differ for iterative methods of different orders,furthermore,they vary for iterative methods of the same order even if they have the same efficiency index.Consequently,this leads to the need for a new index that reflects the real efficiency of the iterative scheme instead of the commonly used efficiency index.