Abstract
In this paper, a new method for solving complex nonlinear systems with symmetric Jacobian matrices is proposed—modified Newton generalized successive overrelaxation (MN-GSOR) iteration method. The new MN-GSOR method helps us to solve nonlinear systems using the GSOR method as an inner iterative approximation for solving the Newton equations. Next, we use the Hölder continuous condition instead of the Lipschitz assumption to analyze and prove the convergence properties of the MN-GSOR method. Numerical experiments are conducted to show the superiority and effectiveness of MN-GSOR method, and comparisons are made to see the advantages over some other recently proposed methods. Furthermore, when the imaginary part of the Jacobian matrix is symmetric but non-definite, some recently proposed methods are not applicable, but the MN-GSOR method still works well.
Similar content being viewed by others
References
Adams LM, LeVeque RJ, Young DM (1988) Analysis of the SOR iteration for the 9-point Laplacian. SIAM J Numer Anal 25(5):1156–1180
Bai Z-Z, Benzi M, Chen F (2010) Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87(3–4):93–111
Bai Z-Z, Benzi M, Chen F (2011) On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer Algorithms 56(2):297–317
Bai Z-Z, Golub GH, Ng MK (2003) Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J Matrix Anal Appl 24(3):603–626
Bai Z-Z, Golub GH, Pan J-Y (2004) Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer Math 98(1):1–32
Bai Z-Z, Guo X-P (2010) On Newton-HSS methods for systems of nonlinear equations with positive definite Jacobian matrices. J Comput Math 28:235–260
Bentrcia A, Zerguine A, Benyoucef M (2010) A reduced complexity chip-level SOR-SIC multiuser detector for long-code CDMA systems. In: 2010 4th international conference on signal processing and communication systems. IEEE, pp 1–4
Cai Y, Zhou Q, Kang, L, Hong X (2008) Sleep transistor sizing for multi-threshold-voltage network using Lagrange SOR iteration. In: 2008 51st midwest symposium on circuits and systems. IEEE, pp. 101–104
Chen M-H, Lin R-F, Wu Q-B (2014) Convergence analysis of the modified Newton-HSS method under the Hölder continuous condition. J Comput Appl Math 264:115–130
Dehghan M, Dehghani-Madiseh M, Hajarian M (2013) A generalized preconditioned MHSS method for a class of complex symmetric linear systems. Math Model Anal 18(4):561–576
Hwang RR, Lin S (1992) On laminar wakes behind a circular cylinder in stratified fluids. J Fluids Eng 114(1):20–28
Karlsson HO (1995) The quasi-minimal residual algorithm applied to complex symmetric linear systems in quantum reactive scattering. J Chem Phys 103(12):4914–4919
Li X, Yang A-L, Wu Y-J (2014) Lopsided PMHSS iteration method for a class of complex symmetric linear systems. Numer Algorithms 66(3):555–568
Ortega JM, Rheinboldt WC (1970) Iterative solution of nonlinear equations in several variables. Academic Press, New York
Papp D, Vizvari B (2006) Effective solution of linear Diophantine equation systems with an application in chemistry. J Math Chem 39(1):15–31
Salkuyeh DK, Hezari D, Edalatpour V (2015) Generalized successive overrelaxation iterative method for a class of complex symmetric linear system of equations. Int J Comput Math 92(4):802–815
Wang J, Guo X-P, Zhong H-X (2018) MN-DPMHSS iteration method for systems of nonlinear equations with block two-by-two complex Jacobian matrices. Numer Algorithms 77(1):167–184
Wu Q-B, Chen M-H (2013) Convergence analysis of modified Newton-HSS method for solving systems of nonlinear equations. Numer Algorithms 64(4):659–683
Yang A-L, Wu Y-J (2012) Newton-MHSS methods for solving systems of nonlinear equations with complex symmetric Jacobian matrices. Numer Algebra Control Optim 2(4):839–853
Zhang Y, Sun Q (2011) Preconditioned bi-conjugate gradient method of large-scale sparse complex linear equation group. Chin J Electron 20(1):192–194
Zheng Q-Q, Ma C-F (2016) Accelerated PMHSS iteration methods for complex symmetric linear systems. Numer Algorithms 73(2):501–516
Zhong H-X, Chen G-L, Guo X-P (2015) On preconditioned modified Newton-MHSS method for systems of nonlinear equations with complex symmetric Jacobian matrices. Numer Algorithms 69(3):553–567
Author information
Authors and Affiliations
Corresponding authors
Additional information
Communicated by Zhong-Zhi Bai.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Qi, X., Wu, HT. & Xiao, XY. Modified Newton-GSOR method for solving complex nonlinear systems with symmetric Jacobian matrices. Comp. Appl. Math. 39, 165 (2020). https://doi.org/10.1007/s40314-020-01204-9
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-020-01204-9
Keywords
- Complex systems of nonlinear equations
- Convergence analysis
- Modified Newton-PMHSS
- Modified Newton-GSOR
- Modified Newton-HSS