Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-28T03:21:49.965Z Has data issue: false hasContentIssue false
  • English
  • Français

A Class of Preconditioned TGHSS-Based Iteration Methods for Weakly Nonlinear Systems

Part of: Numerical linear algebra

Published online by Cambridge University Press:  19 October 2016

Min-Li Zeng  and
Guo-Feng Zhang
Min-Li Zeng*
Affiliation:
School of Mathematics, Putian University, Putian 351100, China
Guo-Feng Zhang*
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
*
*Corresponding author. Email addresses:[email protected] (M.-L. Zeng), [email protected] (G.-F. Zhang)
*Corresponding author. Email addresses:[email protected] (M.-L. Zeng), [email protected] (G.-F. Zhang)
Get access

Abstract

In this paper, we first construct a preconditioned two-parameter generalized Hermitian and skew-Hermitian splitting (PTGHSS) iteration method based on the two-parameter generalized Hermitian and skew-Hermitian splitting (TGHSS) iteration method for non-Hermitian positive definite linear systems. Then a class of PTGHSS-based iteration methods are proposed for solving weakly nonlinear systems based on separable property of the linear and nonlinear terms. The conditions for guaranteeing the local convergence are studied and the quasi-optimal iterative parameters are derived. Numerical experiments are implemented to show that the new methods are feasible and effective for large scale systems of weakly nonlinear systems.

Keywords

System of weakly nonlinear equationsGHSS iteration methodlocal convergenceinner iterationouter iteration

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65F50: Sparse matrices
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 4 , November 2016 , pp. 367 - 383
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aghazadeh, N., Bastani, M. and Salkuyeh, D.K., Generalized Hermitian and skew-Hermitian splitting iterative method for image restoration, Appl. Math. Model., 39 (2015), pp. 6126–6138.Google Scholar
[2] Aghazadeh, N., Khojasteh, S.D. and Bastani, M., Two-parameter generalized Hermitian and skew-Hermitian splitting iteration method, Int. J. Comput. Math., 2015, DOI:10.1080/00207160.2015.1019873.Google Scholar
[3] An, H.-B. and Bai, Z.-Z., NGLM: A globally convergent Newton-GMRES method, Math. Numer. Sinica, 27 (2005), pp. 151–174.Google Scholar
[4] An, H.-B. and Bai, Z.-Z., A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations, Appl. Numer. Math., 57 (2007), pp. 235–252.Google Scholar
[5] Bai, Z.-Z., A class of two-stage iterative methods for systems of weakly nonlinear equations, Numer. Algorithms, 14 (1997), pp. 295–319.Google Scholar
[6] Bai, Z.-Z., Parallel multisplitting two-stage iterative methods for large sparse systems of weakly nonlinear equations, Numer. Algorithms, 15 (1997), pp. 347–372.Google Scholar
[7] Bai, Z.-Z., On the convergence of parallel chaotic nonlinear multisplitting Newton-type methods, J. Comput. Appl. Math., 80 (1997), pp. 317–334.Google Scholar
[8] Bai, Z.-Z., Golub, G.H. and Ng, M.K., Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003), pp. 603–626.Google Scholar
[9] Bai, Z.-Z., Huang, Y.-M. and Ng, M.K., On preconditioned iterative methods for Burgers equations, SIAM J. Sci. Comput., 29 (2007), pp. 415–439.Google Scholar
[10] Bai, Z.-Z. and Yang, X., On HSS-based iteration methods for weakly nonlinear systems, Appl. Numer. Math., 59 (2009), pp. 2923–2936.Google Scholar
[11] Bai, Z.-Z., Optimal parameters in the HSS-like methods for saddle-point problems, Numer. Linear Algebra Appl., 16 (2009), pp. 447–479.Google Scholar
[12] Bai, Z.-Z. and Guo, X.-P., On Newton-HSS methods for systems of nonlinear equations with positive-definite Jacobian matrices, J. Comput. Math., 28 (2010), pp. 235–260.Google Scholar
[13] Benzi, M., A generalization of the Hermitian and skew-Hermitian splitting iteration, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 360–374.Google Scholar
[14] Bertaccini, D., Golub, G.H., Capizzano, S.S. and Possio, C.T., Preconditioned HSS methods for the solution of non-Hermitian positive definite linear systems and applications to the discrete convection-diffusion equation, Numer. Math., 99 (2005), pp. 441–484.Google Scholar
[15] Cao, Y. and Ren, Z.-R., Two variants of the PMHSS iteration method for a class of complex symmetric indefinite linear systems, Appl. Math. Comput., 264 (2015), pp. 61–71.Google Scholar
[16] Elman, H.C. and Golub, G.H., Iterative methods for cyclically reduced nonselfadjoint linear systems, Math. Comp., 54 (1990), pp. 671–700.Google Scholar
[17] Li, X. and Wu, Y.-J., Accelerated Newton-GPSS methods for systems of nonlinear equations, J. Comput. Anal. Appl., 17 (2014), pp. 245–254.Google Scholar
[18] Kelley, C.T., Iterative Methods for Linear and Nonlinear Equations, SIAM, Philadelphia, PA, 1995.Google Scholar
[19] Ortega, J.M. and Rheinboldt, W.C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.Google Scholar
[20] Pour, H.N. and Goughery, H.S., New Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems, Numer. Algorithms, 69 (2015), pp. 207–225.Google Scholar
[21] Pu, Z.-N. and Zhu, M.-Z., A class of iteration methods based on the generalized preconditioned Hermitian and skew-Hermitian splitting for weakly nonlinear systems, J. Comput. Appl. Math., 250 (2013), pp. 16–27.CrossRefGoogle Scholar
[22] Salkuyeh, D.K., The Picard-HSS iteration method for absolute value equations, Optim. Lett., 8 (2014), pp. 2191–2202.Google Scholar
[23] Tang, T., Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, Numer. Math., 61 (1992), pp. 373–382.Google Scholar
[24] Zhang, J.-J., The relaxed nonlinear PHSS-like iterationmethod for absolute value equations, Appl. Math. Comput., 265 (2015), pp. 266–274.Google Scholar
[25] Zhu, M.-Z. and Zhang, G.-F., On CSCS-based iteration methods for Toeplitz system of weakly nonlinear equations, J. Comput. Appl. Math., 235 (2011), pp. 5095–5104.Google Scholar
[26] Zhu, M.-Z. and Zhang, G.-F., A class of iteration methods based on the HSS for Toeplitz systems of weakly nonlinear equations, J. Comput. Appl. Math., 290 (2015), pp. 433–444.Google Scholar