The Hermitian and skew-Hermitian splitting (HSS) method is an unconditionally convergentiteration method for solving large sparse non-Hermitian positive definite system oflinear equations.By making use of the HSS iteration as the inner solver for the Newtonmethod,we establish a class of Newton-HSS methods for solving large sparse systems ofnonlinear equations with positive definite Jacobian matrices at the solution points.For thisclass of inexact Newton methods,two types of local convergence theorems are proved underproper 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 systemsof nonlinear equations arising from the finite difference discretization of a two-dimensionalconvection-diffusion equation perturbed by a nonlinear term.The numerical implementationsalso show that as preconditioners for the Newton-GMRES and the Newton-GCGmethods the HSS iteration outperforms the USOR iteration in both computing time anditeration step.
For large sparse non-Hermitian positive definite system of linear equations,we present several variants of the Hermitian and skew-Hermitian splitting(HSS)about the coefficient matrix and establish correspondingly several HSS-based iterative schemes.Theoretical analyses show that these methods are convergent unconditionally to the exact solution of the referred system of linear equations,and they may show advantages on problems that the HSS method is ineffiective.
BAI Zhong-Zhi State Key Laboratory of Scientific/Engineering Computing,Institute of Computational Mathematics and Scientific/Engineering Computing,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,P.O.Box 2719,Beijing 100080,China
