ON NEWTON-HSS METHODS FOR SYSTEMS OF NONLINEAR EQUATIONS WITH POSITIVE-DEFINITE JACOBIAN MATRICES  

作　　者：Zhong-Zhi Bai Xue-Ping Guo 

机构地区：[1]LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China [2]Department of Mathematics, East China Normal University, Shanghai 200062, China

出　　处：《Journal of Computational Mathematics》2010年第2期235-260,共26页计算数学（英文）

基　　金：supported by the National Basic Research Program (No. 2005CB321702);the China Outstanding Young Scientist Foundation (No. 10525102);the National Natural Science Foundation (No.10471146, No. 10571059 and No. 10571060), P.R. China

摘　　要：The Hermitian and skew-Hermitian splitting （HSS） method is an unconditionally convergent iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By making use of the HSS iteration as the inner solver for the Newton method, we establish a class of Newton-HSS methods for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. For this class of inexact Newton methods, two types of local convergence theorems are proved under proper conditions, and numerical results are given to examine their feasibility and effectiveness. In addition, the advantages of the Newton-HSS methods over the Newton-USOR, the Newton-GMRES and the Newton-GCG methods are shown through solving systems of nonlinear equations arising from the finite difference discretization of a two-dimensional convection-diffusion equation perturbed by a nonlinear term. The numerical implemen- tations also show that as preconditioners for the Newton-GMRES and the Newton-GCG methods the HSS iteration outperforms the USOR iteration in both computing time and iteration step.The Hermitian and skew-Hermitian splitting （HSS） method is an unconditionally convergent iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By making use of the HSS iteration as the inner solver for the Newton method, we establish a class of Newton-HSS methods for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. For this class of inexact Newton methods, two types of local convergence theorems are proved under proper conditions, and numerical results are given to examine their feasibility and effectiveness. In addition, the advantages of the Newton-HSS methods over the Newton-USOR, the Newton-GMRES and the Newton-GCG methods are shown through solving systems of nonlinear equations arising from the finite difference discretization of a two-dimensional convection-diffusion equation perturbed by a nonlinear term. The numerical implemen- tations also show that as preconditioners for the Newton-GMRES and the Newton-GCG methods the HSS iteration outperforms the USOR iteration in both computing time and iteration step.

关 键 词：Systems of nonlinear equations HSS iteration method Newton method Local convergence. 

分 类 号：O241.7[理学—计算数学] O151.21[理学—数学]