Abstract: The Newton iteration method is an important method for solving nonlinear equations. Many other types of iterative methods currently used are based on the Newton iteration method after some extension and expansion. But in these methods, only the properties of the current iteration point and the Jacobi matrix are used, the information about other points and corresponding Jacobi matrices are not fully utilized. In this paper, we use the idea of multiple iterations to improve the Newton iteration method for solving nonlinear equations, and combines the modified Newton iteration method and simplified Newton iteration method to improve the algorithm. We obtain four new types of Newton-type iteration methods for solving nonlinear equations. Rigorous theoretical analyses show that these four Newton-type iterative methods are all convergent. In order to show the effectiveness of proposed algorithms, we present some numerical experimental results. The numerical results show that the four methods all have a fast convergence rate, indicating their efficiency.

Key words: Newton type iteration method, modified Newton method, nonlinear equations, convergence