eng Science and Education Publishing American Journal of Applied Mathematics and Statistics 2328-7292 2017-04-06 5 1 22 32 10.12691/ajams-5-1-5 AJAMS2017515 article Shuping Chen shupingchen@126.com 1 Youhua Qian 2 College of Mathematics, Xiamen University of Technology, xiamen 361024, P. R. China College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, 321004, P. R. China In this article we construct some higher-order modifications of Newton's method for solving nonlinear equations, which is based on the undetermined coefficients. This construction can be applied to any iteration formula. It can be found that per iteration the resulting methods add only one additional function evaluation, their order of convergence can be increased by two or three units. Higher order convergence of our methods is proved and corresponding asymptotic error constants are expressed. Numerical examples, obtained using Matlab with high precision arithmetic, are shown to demonstrate the convergence and efficiency of the combined iterative methods. It is found that the combined iterative methods produce very good results on tested examples, compared to the results produced by the existing higher order schemes in the related literature. http://pubs.sciepub.com/ajams/5/1/5/ajams-5-1-5.pdf Newton's method combined iterative methods nonlinear equations order of convergence computational efficiency