A novel family of composite Newton-Traub methods for solving systems of nonlinear equations

We present a family of three-step iterative methods of convergence order five for solving systems of nonlinear equations. The methodology is based on the two-step Traub's method with cubic convergence for solving scalar equations. Computational efficiency of the new methods is considered and compared with some well-known existing methods. Numerical tests are performed on some problems of different nature, which confirm robust and efficient convergence behavior of the proposed methods. Moreover, theoretical results concerning order of convergence and computational efficiency are verified in the numerical problems. Stability of the methods is tested by drawing basins of attraction in a two-dimensional polynomial system.