Numerical Solution For Systems Of Nonlinear Equations With Structured Complex Jacobian Matrices

In scientific and engineering calculations,some practical problems will be converted into solving problems of nonlinear equations after modeling.For this reason,how to quickly,efficiently,and robustly solve nonlinear equations becomes particularly important.However,for nonlinear equations,it is difficult for general algebraic methods to give exact solutions.Therefore,instead of an exact solution,an approximate solution that satisfies certain precision is used,and the most commonly used method is iteration method.In recent years,for the problem of nonlinear,especially large-scale nonlinear equations,many effective iterative algorithms have been proposed to improve its convergence speed and computational efficiency.However,there are few studies on nonlinear equations with structured complex Jacobian matrix.Therefore,this paper mainly uses extrapolation techniques and combines MN-DPMHSS method to propose a new iterative solution MN-EDPMHSS iterative method.Based on this,combined with multi-step iterative idea,the MMN-EDPMHSS iterative method is proposed.The local convergence of the proposed methods under certain conditions are also discussed.Finally,numerical experiments show that MN-EDPMHSS and MMN-EDPMHSS iterative methods are more effective and feasible than MN-DPMHSS iterative method.In chapter 1,we briefly introduces the related research background and significance,and some basic preliminary knowledge.In chapter 2,based on the extrapolation technique,we propose the EDPMHSS iteration method based on the DPMHSS method proposed in [33].Theoretically,it is proved that under certain conditions,the EDPMHSS iterative method converges faster than the DPMHSS iterative method,and the superiority of the EDPMHSS iterative method over the DPMHSS iterative method can also be seen in numerical experiments in chapter 4.In addition,combining the modified Newton method with the EDPMHSS iteration method,we propose the MN-EDPMHSS iterative method,and give the proof of the local convergence of the MN-EDPMHSS iterative method.In chapter 3,based on the MN-EDPMHSS iterative method,we apply the multi-step iterative technique to propose the MMN-EDPMHSS iterative method and discuss its local convergence.In chapter 4,the numerical experiments show that the MN-EDPMHSS and MMN-EDPM HSS iterative methods proposed by us are more competitive than the MN-DPMHSS iterative method proposed in [33],and the MMN-EDPMHSS iteration method is more efficient than the MN-EDPMHSS iteration.In chapter 5,we give the summary of the thesis and point out how to further our research work.