Multi-step Modified Newton-HSS Methods For Systems Of Nonlinear Equations

In recent years, quite a number of nonlinear systems often arise in many scientific and en-gineering computing areas. The numerical solutions for the nonlinear systems are often required. There are a lot of ways to be chosen to solve the nonlinear systems. In this paper, based on the HSS splitting, we establish a class of multi-step modified Newton-HSS (MMN-HSS) methods for solving large sparse system of nonlinear equations with positive definite Jacobian matrices. The MMN-HSS methods use the multi-step modified Newton methods to solve the nonlinear equation-s, and the HSS method to approximately solve the modified Newton equation. When step number m= 1 and m= 2, this method is simplified as the Newton-HSS method and the modified Newton-HSS method, respectively.Firstly, after some straightforward operations, we give the expression of the multi-step modi-fied Newton-HSS algorithm.Secondly, we analysis the convergence of this method from the following aspects. Under the Lipschitz conditions, we not only show the local convergence theorem but also prove the semilocal convergence theorem. We also give the semilocal convergence theorem of the MMN-HSS meth-ods, assuming the nonlinear operator satisfies the Holder continuous condition. Then, we establish the global multi-step modified Newton-HSS method and prove the global convergence theorem. The Holder condition is much milder than the usual Lipschitz condition. To some degree, the Lip-schitz condition is the special example of the Holder condition.Finally, one numerical example with the Lipschitz condition and two examples with the Holder condition are given to confirm the feasibility and effectiveness of our method. The nu-merical results show that the multi-step modified Newton-HSS method outperforms the modified Newton-HSS method in the sense of numbers of iterations and CPU time.