Abstract
It has been proved that Newton–HSS method is efficient and robust for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. In this paper, by utilizing the single-step Hermitian and skew-Hermitian splitting (SHSS) iteration technique, which performs efficiently under certain conditions, as the inner solver of the modified Newton method, we propose a class of modified Newton–SHSS methods. Subsequently, the local and semilocal convergence properties of our method will be discussed under some reasonable assumptions. Furthermore, we introduce the modified Newton–SHSS method with a backtracking strategy and analyze its basic global convergence theorem. Finally, several typical instances are used to illustrate the advantages of our methods when the Hermitian part of the Jacobian matrices are dominant.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amiri A, Cordero A, Darvishi MT, Torregrosa JR (2018) A fast algorithm to solve systems of nonlinear equations. J Comput Appl Math. https://doi.org/10.1016/j.cam.2018.03.048
Bai ZZ, Golub GH, NG MK (2003) Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J Matrix Anal Appl 24:603–626
Bai ZZ, Golub GH, Li CK (2006) Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices, vol 28. Society for Industrial and Applied Mathematics, Philadelphia, pp 583–603
Bai ZZ, Golub GH, Li CK (2007) Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices. Math Comput 76:287–298
Bai ZZ, Golub GH, Ng MK (2008) On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. Linear Algebra Appl 428:413–440
Bai ZZ, Guo XP (2010) On Newton–HSS methods for systems of nonlinear equations with positive-definite Jacobian matrices. J Comput Math 28:235–260
Bai ZZ, Benzi M, Chen F (2010) Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87:93–111
Brown PN, Saad Y (1990) Hybrid Krylov methods for nonlinear systems of equations. SIAM J Sci Stat Comput 11:450–481
Brown PN, Saad Y (1994) Convergence theory of nonlinear Newton–Krylov algorithms. SIAM J Optim 4:297–330
Chen MH, Lin RF, Wu QB (2014) Convergence analysis of the modified Newton–HSS method under the Hlder continuous condition. J Comput Appl Math 264:115–130
Dai PF, Wu QB, Chen MH (2018) Modified Newton–NSS method for solving systems of nonlinear equations. Numer Algorithms 77:1–21
Darvishi MT, Barati A (2007) A third-order Newton-type method to solve systems of nonlinear equations. Appl Math Comput 187:630–635
Dembo RS, Eisenstat SC, Steihaug T (1982) Inexact Newton methods. SIAM J Numer Anal 19:400–408
Guo XP (2007) On semilocal convergence of inexact Newton methods. J Comput Math 25:231–242
Guo XP, Duff IS (2011) Semilocal and global convergence of the Newton–HSS method for systems of nonlinear equations. Numer Linear Algebra Appl 18:299–315
Huang ZD (2003) Convergence of inexact Newton method. J Zhejiang Univ Sci 4:393–396
Kelley CT (1995) Iterative methods for linear and nonlinear equations. Society for Industrial and Applied Mathematics, Philadelphia
Li Y, Guo XP (2017) Multi-step modified Newton–HSS methods for systems of nonlinear equations with positive definite Jacobian matrices. Numer Algorithms 75:55–80
Li X, Wu YJ (2014) Accelerated Newton–GPSS methods for systems of nonlinear equations. J Comput Anal Appl 17:245–254
Li CX, Wu SL (2015) A single-step HSS method for non-Hermitian positive definite linear systems. Appl Math Lett 44:26–29
Ortega JM, Rheinboldt WC (1970) Iterative solution of nonlinear equations in several variables. Academic Press, Cambridge
Pernice M, Walker HF (1998) NITSOL: a Newton iterative solver for nonlinear systems, vol 19. Society for Industrial and Applied Mathematics, Philadelphia, pp 302–318
Saad Y (1996) Iterative methods for sparse linear systems. Society for Industrial and Applied Mathematics, Philadelphia
Wu QB, Chen MH (2013) Convergence analysis of modified Newton–HSS method for solving systems of nonlinear equations. Numer Algorithms 64:659–683
Wang J, Guo XP, Zhong HX (2018) MN-DPMHSS iteration method for systems of nonlinear equations with block two-by-two complex Jacobian matrices. Numer Algorithms 77:167–184
Wu YJ, Li X, Yuan JY (2017) A non-alternating preconditioned HSS iteration method for non-Hermitian positive definite linear systems. Comput Appl Math 36:367–381
Yang AL, Wu YJ (2013) Newton–MHSS methods for solving systems of nonlinear equations with complex symmetric Jacobian matrices. Numer Algebra Control Optim 2:839–853
Young DM (1971) Iterative solution of large linear systems. Academic Press, New York
Zhong HX, Chen GL, Guo XP (2015) On preconditioned modified Newton–MHSS method for systems of nonlinear equations with complex symmetric Jacobian matrices. Numer Algorithms 69:553–567
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11771393, 11632015) and Zhejiang Natural Science Foundation (Grant No. LZ14A010002).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Andreas Fischer.
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
Xie, F., Wu, QB. & Dai, PF. Modified Newton–SHSS method for a class of systems of nonlinear equations. Comp. Appl. Math. 38, 19 (2019). https://doi.org/10.1007/s40314-019-0793-9
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-019-0793-9
Keywords
- Nonlinear systems
- Single-step Hermitian and skew-Hermitian splitting
- Modified Newton method
- Convergence properties