Newton-MHSS methods for solving systems of nonlinear equations with complex symmetric Jacobian matrices

Previous Article
A new Bramble-Pasciak-like preconditioner for saddle point problems
NACO Home
This Issue
Next Article
On convergence of the inner-outer iteration method for computing PageRank

2012, 2(4): 839-853. doi: 10.3934/naco.2012.2.839

Newton-MHSS methods for solving systems of nonlinear equations with complex symmetric Jacobian matrices

1.	School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China, China

Received December 2011 Revised August 2012 Published November 2012

Abstract
Related Papers

Modified Hermitian and skew-Hermitian splitting (MHSS) method is an unconditionally convergent iterative method for solving large sparse complex symmetric systems of linear equations. By making use of the MHSS iteration as the inner solver for the inexact Newton method, we establish a class of inexact Newton-MHSS methods for solving large sparse systems of nonlinear equations with complex symmetric Jacobian matrices at the solution points. The local and semi-local convergence properties are analyzed under some proper assumptions. Moreover, by introducing a backtracking linear search technique, a kind of global convergence inexact Newton-MHSS methods are also presented and analyzed. Numerical results are given to examine the feasibility and effectiveness of the inexact Newton-MHSS methods.

Keywords: MHSS method, inexact Newton method, convergence property., Nonlinear equations, complex linear system.

Mathematics Subject Classification: Primary: 65H10,65F10; Secondary: 65H2.

Citation: Ai-Li Yang, Yu-Jiang Wu. Newton-MHSS methods for solving systems of nonlinear equations with complex symmetric Jacobian matrices. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 839-853. doi: 10.3934/naco.2012.2.839

References:

[1]	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), 235. doi: 10.1016/j.apnum.2006.02.007. Google Scholar
[2]	I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation,, Rev. Mod. Phys., 74 (2002), 99. doi: 10.1103/RevModPhys.74.99. Google Scholar
[3]	Z. Z. Bai, On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems,, Computing, 89 (2010), 171. Google Scholar
[4]	Z. Z. Bai, M. Benzi and F. Chen, Modified HSS iteration methods for a class of complex symmetric linear systems,, Computing, 87 (2010), 93. doi: 10.1007/s00607-010-0077-0. Google Scholar
[5]	Z. Z. Bai, M. Benzi and F. Chen, On preconditioned MHSS iteration methods for complex symmetric linear systems,, Numer. Algor., 56 (2011), 297. doi: 10.1007/s11075-010-9441-6. Google Scholar
[6]	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 (2002), 603. doi: 10.1137/S0895479801395458. Google Scholar
[7]	Z.-Z. Bai and X.-P. Guo, On Newton-HSS methods for systems of nonlinear equations with positive-definite jacobian matrices,, J. Comput. Math., 28 (2010), 235. Google Scholar
[8]	Z. Z. Bai and X. Yang, On HSS-based iteration methods for weakly nonlinear systems,, Appl. Numer. Math., 59 (2009), 2923. doi: 10.1016/j.apnum.2009.06.005. Google Scholar
[9]	M. Benzi and D. B. Szyld, Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods,, Numer. Math., 76 (1997), 309. doi: 10.1007/s002110050265. Google Scholar
[10]	T. Bohr, M. H. Jensen, G. Paladin and A. Vulpiani, "Dynamical Systems Approach to Turbulence,", Cambridge University Press, (1998). doi: 10.1017/CBO9780511599972. Google Scholar
[11]	R. Dembo, S. Eisenstat and T. Steihaug, Inexact Newton methods,, SIAM J. Numer. Anal., 19 (1982), 400. doi: 10.1137/0719025. Google Scholar
[12]	P. Deuflhard, "Newton Methods for Nonlinear Problems,", Springer-Verlag, (2004). Google Scholar
[13]	S. C. Eisenstat and H. F. Walker, Globally convergent inexact Newton methods,, SIAM J. Optim., 4 (1994), 393. doi: 10.1137/0804022. Google Scholar
[14]	S. C. Eisenstat and H. F. Walker, Choosing the forcing terms in an inexact Newton method,, SIAM J. Sci. Comput., 17 (1996), 16. doi: 10.1137/0917003. Google Scholar
[15]	X. P. Guo and I. S. Duff, Semilocal and global convergence of the Newton-HSS method for systems of nonlinear equations,, Numer. Linear Algebra Appl., 18 (2011), 299. doi: 10.1002/nla.713. Google Scholar
[16]	C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations,", SIAM, (1995). doi: 10.1137/1.9781611970944. Google Scholar
[17]	Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence,", Dover Publications, (2003). Google Scholar
[18]	J. M. Ortega and W. C. Rheinboldt, "Iterative Solution of Nonlinear Equations in Several Variables,", SIAM, (2000). doi: 10.1137/1.9780898719468. Google Scholar
[19]	M. Pernice and H. F. Walker, Nitsol: A newton iterative solver for nonlinear systems,, SIAM J. Sci. Comput., 19 (1998), 302. doi: 10.1137/S1064827596303843. Google Scholar
[20]	Y. Saad, "Iterative Methods for Sparse Linear Systems,", 2^nd edition, (2003). doi: 10.1137/1.9780898718003. Google Scholar
[21]	C. Sulem and P. L. Sulem, "The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse,", Springer Verlag, (1999). Google Scholar
[22]	A. L. Yang, J. An and Y. J. Wu, A generalized preconditioned HSS method for non-Hermitian positive definite linear systems,, Appl. Math. Comput., 216 (2010), 1715. doi: 10.1016/j.amc.2009.12.032. Google Scholar

show all references

References:

[1]	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), 235. doi: 10.1016/j.apnum.2006.02.007. Google Scholar
[2]	I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation,, Rev. Mod. Phys., 74 (2002), 99. doi: 10.1103/RevModPhys.74.99. Google Scholar
[3]	Z. Z. Bai, On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems,, Computing, 89 (2010), 171. Google Scholar
[4]	Z. Z. Bai, M. Benzi and F. Chen, Modified HSS iteration methods for a class of complex symmetric linear systems,, Computing, 87 (2010), 93. doi: 10.1007/s00607-010-0077-0. Google Scholar
[5]	Z. Z. Bai, M. Benzi and F. Chen, On preconditioned MHSS iteration methods for complex symmetric linear systems,, Numer. Algor., 56 (2011), 297. doi: 10.1007/s11075-010-9441-6. Google Scholar
[6]	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 (2002), 603. doi: 10.1137/S0895479801395458. Google Scholar
[7]	Z.-Z. Bai and X.-P. Guo, On Newton-HSS methods for systems of nonlinear equations with positive-definite jacobian matrices,, J. Comput. Math., 28 (2010), 235. Google Scholar
[8]	Z. Z. Bai and X. Yang, On HSS-based iteration methods for weakly nonlinear systems,, Appl. Numer. Math., 59 (2009), 2923. doi: 10.1016/j.apnum.2009.06.005. Google Scholar
[9]	M. Benzi and D. B. Szyld, Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods,, Numer. Math., 76 (1997), 309. doi: 10.1007/s002110050265. Google Scholar
[10]	T. Bohr, M. H. Jensen, G. Paladin and A. Vulpiani, "Dynamical Systems Approach to Turbulence,", Cambridge University Press, (1998). doi: 10.1017/CBO9780511599972. Google Scholar
[11]	R. Dembo, S. Eisenstat and T. Steihaug, Inexact Newton methods,, SIAM J. Numer. Anal., 19 (1982), 400. doi: 10.1137/0719025. Google Scholar
[12]	P. Deuflhard, "Newton Methods for Nonlinear Problems,", Springer-Verlag, (2004). Google Scholar
[13]	S. C. Eisenstat and H. F. Walker, Globally convergent inexact Newton methods,, SIAM J. Optim., 4 (1994), 393. doi: 10.1137/0804022. Google Scholar
[14]	S. C. Eisenstat and H. F. Walker, Choosing the forcing terms in an inexact Newton method,, SIAM J. Sci. Comput., 17 (1996), 16. doi: 10.1137/0917003. Google Scholar
[15]	X. P. Guo and I. S. Duff, Semilocal and global convergence of the Newton-HSS method for systems of nonlinear equations,, Numer. Linear Algebra Appl., 18 (2011), 299. doi: 10.1002/nla.713. Google Scholar
[16]	C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations,", SIAM, (1995). doi: 10.1137/1.9781611970944. Google Scholar
[17]	Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence,", Dover Publications, (2003). Google Scholar
[18]	J. M. Ortega and W. C. Rheinboldt, "Iterative Solution of Nonlinear Equations in Several Variables,", SIAM, (2000). doi: 10.1137/1.9780898719468. Google Scholar
[19]	M. Pernice and H. F. Walker, Nitsol: A newton iterative solver for nonlinear systems,, SIAM J. Sci. Comput., 19 (1998), 302. doi: 10.1137/S1064827596303843. Google Scholar
[20]	Y. Saad, "Iterative Methods for Sparse Linear Systems,", 2^nd edition, (2003). doi: 10.1137/1.9780898718003. Google Scholar
[21]	C. Sulem and P. L. Sulem, "The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse,", Springer Verlag, (1999). Google Scholar
[22]	A. L. Yang, J. An and Y. J. Wu, A generalized preconditioned HSS method for non-Hermitian positive definite linear systems,, Appl. Math. Comput., 216 (2010), 1715. doi: 10.1016/j.amc.2009.12.032. Google Scholar

[1]	Jinyan Fan, Jianyu Pan. Inexact Levenberg-Marquardt method for nonlinear equations. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1223-1232. doi: 10.3934/dcdsb.2004.4.1223
[2]	Tahereh Salimi Siahkolaei, Davod Khojasteh Salkuyeh. A preconditioned SSOR iteration method for solving complex symmetric system of linear equations. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019033
[3]	Jinyan Fan, Jianyu Pan. On the convergence rate of the inexact Levenberg-Marquardt method. Journal of Industrial & Management Optimization, 2011, 7 (1) : 199-210. doi: 10.3934/jimo.2011.7.199
[4]	Shuang Chen, Li-Ping Pang, Dan Li. An inexact semismooth Newton method for variational inequality with symmetric cone constraints. Journal of Industrial & Management Optimization, 2015, 11 (3) : 733-746. doi: 10.3934/jimo.2015.11.733
[5]	Saeed Ketabchi, Hossein Moosaei, M. Parandegan, Hamidreza Navidi. Computing minimum norm solution of linear systems of equations by the generalized Newton method. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 113-119. doi: 10.3934/naco.2017008
[6]	Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013
[7]	Wei-Zhe Gu, Li-Yong Lu. The linear convergence of a derivative-free descent method for nonlinear complementarity problems. Journal of Industrial & Management Optimization, 2017, 13 (2) : 531-548. doi: 10.3934/jimo.2016030
[8]	Henryk Leszczyński, Monika Wrzosek. Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion. Mathematical Biosciences & Engineering, 2017, 14 (1) : 237-248. doi: 10.3934/mbe.2017015
[9]	C.Y. Wang, M.X. Li. Convergence property of the Fletcher-Reeves conjugate gradient method with errors. Journal of Industrial & Management Optimization, 2005, 1 (2) : 193-200. doi: 10.3934/jimo.2005.1.193
[10]	Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic & Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1
[11]	Xiaojiao Tong, Shuzi Zhou. A smoothing projected Newton-type method for semismooth equations with bound constraints. Journal of Industrial & Management Optimization, 2005, 1 (2) : 235-250. doi: 10.3934/jimo.2005.1.235
[12]	Hongxiu Zhong, Guoliang Chen, Xueping Guo. Semi-local convergence of the Newton-HSS method under the center Lipschitz condition. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 85-99. doi: 10.3934/naco.2019007
[13]	Yu-Ning Yang, Su Zhang. On linear convergence of projected gradient method for a class of affine rank minimization problems. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1507-1519. doi: 10.3934/jimo.2016.12.1507
[14]	Dan Li, Li-Ping Pang, Fang-Fang Guo, Zun-Quan Xia. An alternating linearization method with inexact data for bilevel nonsmooth convex optimization. Journal of Industrial & Management Optimization, 2014, 10 (3) : 859-869. doi: 10.3934/jimo.2014.10.859
[15]	T. Tachim Medjo. On the Newton method in robust control of fluid flow. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1201-1222. doi: 10.3934/dcds.2003.9.1201
[16]	Xiaojiao Tong, Felix F. Wu, Yongping Zhang, Zheng Yan, Yixin Ni. A semismooth Newton method for solving optimal power flow. Journal of Industrial & Management Optimization, 2007, 3 (3) : 553-567. doi: 10.3934/jimo.2007.3.553
[17]	Zhi-Feng Pang, Yu-Fei Yang. Semismooth Newton method for minimization of the LLT model. Inverse Problems & Imaging, 2009, 3 (4) : 677-691. doi: 10.3934/ipi.2009.3.677
[18]	Matthias Gerdts, Stefan Horn, Sven-Joachim Kimmerle. Line search globalization of a semismooth Newton method for operator equations in Hilbert spaces with applications in optimal control. Journal of Industrial & Management Optimization, 2017, 13 (1) : 47-62. doi: 10.3934/jimo.2016003
[19]	Zhiyou Wu, Fusheng Bai, Guoquan Li, Yongjian Yang. A new auxiliary function method for systems of nonlinear equations. Journal of Industrial & Management Optimization, 2015, 11 (2) : 345-364. doi: 10.3934/jimo.2015.11.345
[20]	Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59

Impact Factor:

American Institute of Mathematical Sciences