Efficient derivative-free numerical methods for solving systems of nonlinear equations

We present iterative methods of convergence order four and six for solving systems of nonlinear equations. The algorithms are free from any derivative evaluations per full iteration. First-order divided difference operator for functions of several variables and direct computation by Taylor's expansion are used to prove the local convergence order. Computational efficiency is discussed and the comparison between efficiencies of proposed techniques with existing derivative-free techniques is performed. It is shown that the new methods are especially efficient in solving large systems. Numerical tests are performed on some problems of different nature, which confirm robust and efficient convergence behavior of the proposed methods. Moreover, theoretical results are also verified in the examples.