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

  • Min-Li Zeng (a1) and Guo-Feng Zhang (a2)
Abstract
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.

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Corresponding author
*Corresponding author. Email addresses:zengml12@lzu.edu.cn (M.-L. Zeng), gf_zhang@lzu.edu.cn (G.-F. Zhang)
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[1] N.Aghazadeh , M.Bastani and D.K.Salkuyeh , Generalized Hermitian and skew-Hermitian splitting iterative method for image restoration, Appl. Math. Model., 39 (2015), pp. 61266138.

[4] H.-B.An and Z.-Z.Bai , A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations, Appl. Numer. Math., 57 (2007), pp. 235252.

[5] Z.-Z.Bai , A class of two-stage iterative methods for systems of weakly nonlinear equations, Numer. Algorithms, 14 (1997), pp. 295319.

[6] Z.-Z.Bai , Parallel multisplitting two-stage iterative methods for large sparse systems of weakly nonlinear equations, Numer. Algorithms, 15 (1997), pp. 347372.

[7] Z.-Z.Bai , On the convergence of parallel chaotic nonlinear multisplitting Newton-type methods, J. Comput. Appl. Math., 80 (1997), pp. 317334.

[8] Z.-Z.Bai , G.H.Golub and M.K.Ng , Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003), pp. 603626.

[9] Z.-Z.Bai , Y.-M.Huang and M.K.Ng , On preconditioned iterative methods for Burgers equations, SIAM J. Sci. Comput., 29 (2007), pp. 415439.

[10] Z.-Z.Bai and X.Yang , On HSS-based iteration methods for weakly nonlinear systems, Appl. Numer. Math., 59 (2009), pp. 29232936.

[11] Z.-Z.Bai , Optimal parameters in the HSS-like methods for saddle-point problems, Numer. Linear Algebra Appl., 16 (2009), pp. 447479.

[13] M.Benzi , A generalization of the Hermitian and skew-Hermitian splitting iteration, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 360374.

[14] D.Bertaccini , G.H.Golub , S.S.Capizzano and C.T.Possio , 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. 441484.

[15] Y.Cao and Z.-R.Ren , Two variants of the PMHSS iteration method for a class of complex symmetric indefinite linear systems, Appl. Math. Comput., 264 (2015), pp. 6171.

[16] H.C.Elman and G.H.Golub , Iterative methods for cyclically reduced nonselfadjoint linear systems, Math. Comp., 54 (1990), pp. 671700.

[18] C.T.Kelley , Iterative Methods for Linear and Nonlinear Equations, SIAM, Philadelphia, PA, 1995.

[19] J.M.Ortega and W.C.Rheinboldt , Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.

[20] H.N.Pour and H.S.Goughery , New Hermitian and skew-Hermitian splitting methods for non-Hermitian positive-definite linear systems, Numer. Algorithms, 69 (2015), pp. 207225.

[21] Z.-N.Pu and M.-Z.Zhu , 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. 1627.

[22] D.K.Salkuyeh , The Picard-HSS iteration method for absolute value equations, Optim. Lett., 8 (2014), pp. 21912202.

[23] T.Tang , Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, Numer. Math., 61 (1992), pp. 373382.

[24] J.-J.Zhang , The relaxed nonlinear PHSS-like iterationmethod for absolute value equations, Appl. Math. Comput., 265 (2015), pp. 266274.

[25] M.-Z.Zhu and G.-F.Zhang , On CSCS-based iteration methods for Toeplitz system of weakly nonlinear equations, J. Comput. Appl. Math., 235 (2011), pp. 50955104.

[26] M.-Z.Zhu and G.-F.Zhang , A class of iteration methods based on the HSS for Toeplitz systems of weakly nonlinear equations, J. Comput. Appl. Math., 290 (2015), pp. 433444.

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East Asian Journal on Applied Mathematics
  • ISSN: 2079-7362
  • EISSN: 2079-7370
  • URL: /core/journals/east-asian-journal-on-applied-mathematics
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