Volume 28, Issue 2
On Newton-HSS Methods for Systems of Nonlinear Equations with Positive-Definite Jacobian Matrices

Zhong-Zhi Bai & Xue-Ping Guo

J. Comp. Math., 28 (2010), pp. 235-260.

Published online: 2010-04

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  • Abstract

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 implementations 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.

  • Keywords

Systems of nonlinear equations, HSS iteration method, Newton method, Local convergence.

  • AMS Subject Headings

65F10, 65W05.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-28-235, author = {}, title = {On Newton-HSS Methods for Systems of Nonlinear Equations with Positive-Definite Jacobian Matrices}, journal = {Journal of Computational Mathematics}, year = {2010}, volume = {28}, number = {2}, pages = {235--260}, abstract = {

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 implementations 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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2009.10-m2836}, url = {http://global-sci.org/intro/article_detail/jcm/8517.html} }
TY - JOUR T1 - On Newton-HSS Methods for Systems of Nonlinear Equations with Positive-Definite Jacobian Matrices JO - Journal of Computational Mathematics VL - 2 SP - 235 EP - 260 PY - 2010 DA - 2010/04 SN - 28 DO - http://doi.org/10.4208/jcm.2009.10-m2836 UR - https://global-sci.org/intro/article_detail/jcm/8517.html KW - Systems of nonlinear equations, HSS iteration method, Newton method, Local convergence. AB -

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 implementations 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.

Zhong-Zhi Bai & Xue-Ping Guo. (1970). On Newton-HSS Methods for Systems of Nonlinear Equations with Positive-Definite Jacobian Matrices. Journal of Computational Mathematics. 28 (2). 235-260. doi:10.4208/jcm.2009.10-m2836
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