Accelerated Modified Newton-HSS Methods For Systems Of Nonlinear Equations

In recent years,nonlinear problems are increasingly emerging in the fields of science and engineering computing.How to improve the efficiency of the solution has been widely recognized.In this thesis we mainly discuss the accelerated modified Newton-HSS(AMN-HSS)methods,which are used for solving large sparse systems of nonlinear equations with non Hermitian Jaco-bian matrix.The AMN-HSS methods are based on the HSS splitting,and combine the accelerated modified Newton methods to solve the nonlinear systems and the HSS method to approximately solve the Newton equations.When the step size parameters a = 1 and b= 1,we can get the known modified Newton-HSS method.This thesis is organized as follows.Firstly,we give the accelerated modified Newton-HSS algorithm 1 for solving systems of nonlinear equations with non-Hermitian positive definite Jacobian matrices,and the accelerated modified Newton-HSS algorithm 2 for solving systems of nonlinear equations with non-Hermitian singular Jacobian matrices.Then their convergence analysises are showed from the following as-pects:1.The local convergence of the accelerated modified Newton-HSS(AMN-HSS 1)method 1 is proved under the Lipschitz conditions.2.The semilocal convergence of the accelerated modified Newton-HSS method 1(AMN-HSS 1)is proved under the Lipschitz conditions.3.The local convergence of the accelerated modified Newton-HSS method 2(AMN-HSS 2)is proved under the Lipschitz conditions.Finally,the numerical example 6.1,under the Lipschitz condition,with a= 1,b being the optimal parameters and b = 1,a being optimal parameter,illustrates the accelerated modified Newton-HSS method 1 is better than the modified Newton-HSS method in respect of numbers of the inner iterations and CPU time.Then,the efficiency of the accelerated modified Newton-HSS method 2 are illustrated through the numerical example 6.2 under the Lipschitz condition.Thus,we demonstrate the feasibility and validity of the accelerated modified Newton-HSS algorithms.