A simple and efficient method with high order convergence for solving systems of nonlinear equations

We propose an m+1-step modified Newton method of convergence order m+2 to solve systems of nonlinear equations which are third Fréchet differentiable in a convex set containing the zero. Computational efficiency in the general form for a positive integer m is discussed, which shows that the efficiency increases with m when applied to large systems. Moreover, a comparison between the efficiency of this technique and some existing efficient methods is made, which implies that the present method is more efficient particularly for solving large systems of equations. Theoretical results about order of convergence and computational efficiency are largely verified in numerical examples.