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

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.
Citation: Ai-Li Yang, Yu-Jiang Wu. Newton-MHSS methods for solving systems of nonlinear equations with complex symmetric Jacobian matrices. Numerical Algebra, Control and 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), 235-252. doi: 10.1016/j.apnum.2006.02.007.

[2]

I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation, Rev. Mod. Phys., 74 (2002), 99-143. doi: 10.1103/RevModPhys.74.99.

[3]

Z. Z. Bai, On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems, Computing, 89 (2010), 171-197.

[4]

Z. Z. Bai, M. Benzi and F. Chen, Modified HSS iteration methods for a class of complex symmetric linear systems, Computing, 87 (2010), 93-111. doi: 10.1007/s00607-010-0077-0.

[5]

Z. Z. Bai, M. Benzi and F. Chen, On preconditioned MHSS iteration methods for complex symmetric linear systems, Numer. Algor., 56 (2011), 297-317. doi: 10.1007/s11075-010-9441-6.

[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), 603-626. doi: 10.1137/S0895479801395458.

[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), 235-260.

[8]

Z. Z. Bai and X. Yang, On HSS-based iteration methods for weakly nonlinear systems, Appl. Numer. Math., 59 (2009), 2923-2936. doi: 10.1016/j.apnum.2009.06.005.

[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), 309-321. doi: 10.1007/s002110050265.

[10]

T. Bohr, M. H. Jensen, G. Paladin and A. Vulpiani, "Dynamical Systems Approach to Turbulence," Cambridge University Press, 1998. doi: 10.1017/CBO9780511599972.

[11]

R. Dembo, S. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982), 400-408. doi: 10.1137/0719025.

[12]

P. Deuflhard, "Newton Methods for Nonlinear Problems," Springer-Verlag, Berlin Heidelberg, 2004.

[13]

S. C. Eisenstat and H. F. Walker, Globally convergent inexact Newton methods, SIAM J. Optim., 4 (1994), 393-422. doi: 10.1137/0804022.

[14]

S. C. Eisenstat and H. F. Walker, Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput., 17 (1996), 16-32. doi: 10.1137/0917003.

[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), 299-315. doi: 10.1002/nla.713.

[16]

C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations," SIAM, Philadelphia, PA, 1995. doi: 10.1137/1.9781611970944.

[17]

Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence," Dover Publications, Inc., Mineola, New York, 2003.

[18]

J. M. Ortega and W. C. Rheinboldt, "Iterative Solution of Nonlinear Equations in Several Variables," SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719468.

[19]

M. Pernice and H. F. Walker, Nitsol: A newton iterative solver for nonlinear systems, SIAM J. Sci. Comput., 19 (1998), 302-318. doi: 10.1137/S1064827596303843.

[20]

Y. Saad, "Iterative Methods for Sparse Linear Systems," 2nd edition, SIAM, Philadelphia, PA, 2003. doi: 10.1137/1.9780898718003.

[21]

C. Sulem and P. L. Sulem, "The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse," Springer Verlag, New York, 1999.

[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), 1715-1722. doi: 10.1016/j.amc.2009.12.032.

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), 235-252. doi: 10.1016/j.apnum.2006.02.007.

[2]

I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation, Rev. Mod. Phys., 74 (2002), 99-143. doi: 10.1103/RevModPhys.74.99.

[3]

Z. Z. Bai, On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems, Computing, 89 (2010), 171-197.

[4]

Z. Z. Bai, M. Benzi and F. Chen, Modified HSS iteration methods for a class of complex symmetric linear systems, Computing, 87 (2010), 93-111. doi: 10.1007/s00607-010-0077-0.

[5]

Z. Z. Bai, M. Benzi and F. Chen, On preconditioned MHSS iteration methods for complex symmetric linear systems, Numer. Algor., 56 (2011), 297-317. doi: 10.1007/s11075-010-9441-6.

[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), 603-626. doi: 10.1137/S0895479801395458.

[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), 235-260.

[8]

Z. Z. Bai and X. Yang, On HSS-based iteration methods for weakly nonlinear systems, Appl. Numer. Math., 59 (2009), 2923-2936. doi: 10.1016/j.apnum.2009.06.005.

[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), 309-321. doi: 10.1007/s002110050265.

[10]

T. Bohr, M. H. Jensen, G. Paladin and A. Vulpiani, "Dynamical Systems Approach to Turbulence," Cambridge University Press, 1998. doi: 10.1017/CBO9780511599972.

[11]

R. Dembo, S. Eisenstat and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal., 19 (1982), 400-408. doi: 10.1137/0719025.

[12]

P. Deuflhard, "Newton Methods for Nonlinear Problems," Springer-Verlag, Berlin Heidelberg, 2004.

[13]

S. C. Eisenstat and H. F. Walker, Globally convergent inexact Newton methods, SIAM J. Optim., 4 (1994), 393-422. doi: 10.1137/0804022.

[14]

S. C. Eisenstat and H. F. Walker, Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput., 17 (1996), 16-32. doi: 10.1137/0917003.

[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), 299-315. doi: 10.1002/nla.713.

[16]

C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations," SIAM, Philadelphia, PA, 1995. doi: 10.1137/1.9781611970944.

[17]

Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence," Dover Publications, Inc., Mineola, New York, 2003.

[18]

J. M. Ortega and W. C. Rheinboldt, "Iterative Solution of Nonlinear Equations in Several Variables," SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719468.

[19]

M. Pernice and H. F. Walker, Nitsol: A newton iterative solver for nonlinear systems, SIAM J. Sci. Comput., 19 (1998), 302-318. doi: 10.1137/S1064827596303843.

[20]

Y. Saad, "Iterative Methods for Sparse Linear Systems," 2nd edition, SIAM, Philadelphia, PA, 2003. doi: 10.1137/1.9780898718003.

[21]

C. Sulem and P. L. Sulem, "The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse," Springer Verlag, New York, 1999.

[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), 1715-1722. doi: 10.1016/j.amc.2009.12.032.

[1]

Jinyan Fan, Jianyu Pan. Inexact Levenberg-Marquardt method for nonlinear equations. Discrete and 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 and Optimization, 2019, 9 (4) : 483-492. doi: 10.3934/naco.2019033

[3]

Jinyan Fan, Jianyu Pan. On the convergence rate of the inexact Levenberg-Marquardt method. Journal of Industrial and 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 and Management Optimization, 2015, 11 (3) : 733-746. doi: 10.3934/jimo.2015.11.733

[5]

Hong-Yi Miao, Li Wang. Preconditioned inexact Newton-like method for large nonsymmetric eigenvalue problems. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 677-685. doi: 10.3934/naco.2021012

[6]

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 and Optimization, 2017, 7 (2) : 113-119. doi: 10.3934/naco.2017008

[7]

Jahnabi Chakravarty, Ashiho Athikho, Manideepa Saha. Convergence of interval AOR method for linear interval equations. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 293-308. doi: 10.3934/naco.2021006

[8]

Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial and Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013

[9]

Wei-Zhe Gu, Li-Yong Lu. The linear convergence of a derivative-free descent method for nonlinear complementarity problems. Journal of Industrial and Management Optimization, 2017, 13 (2) : 531-548. doi: 10.3934/jimo.2016030

[10]

Leyu Hu, Xingju Cai. Convergence of a randomized Douglas-Rachford method for linear system. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 463-474. doi: 10.3934/naco.2020045

[11]

Jie-Wen He, Chi-Chon Lei, Chen-Yang Shi, Seak-Weng Vong. An inexact alternating direction method of multipliers for a kind of nonlinear complementarity problems. Numerical Algebra, Control and Optimization, 2021, 11 (3) : 353-362. doi: 10.3934/naco.2020030

[12]

Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185

[13]

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

[14]

Haiyan Wang, Jinyan Fan. Convergence properties of inexact Levenberg-Marquardt method under Hölderian local error bound. Journal of Industrial and Management Optimization, 2021, 17 (4) : 2265-2275. doi: 10.3934/jimo.2020068

[15]

C.Y. Wang, M.X. Li. Convergence property of the Fletcher-Reeves conjugate gradient method with errors. Journal of Industrial and Management Optimization, 2005, 1 (2) : 193-200. doi: 10.3934/jimo.2005.1.193

[16]

Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic and Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1

[17]

Xiaojiao Tong, Shuzi Zhou. A smoothing projected Newton-type method for semismooth equations with bound constraints. Journal of Industrial and Management Optimization, 2005, 1 (2) : 235-250. doi: 10.3934/jimo.2005.1.235

[18]

Hongxiu Zhong, Guoliang Chen, Xueping Guo. Semi-local convergence of the Newton-HSS method under the center Lipschitz condition. Numerical Algebra, Control and Optimization, 2019, 9 (1) : 85-99. doi: 10.3934/naco.2019007

[19]

Wei Ouyang, Li Li. Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods. Journal of Industrial and Management Optimization, 2021, 17 (1) : 169-184. doi: 10.3934/jimo.2019105

[20]

Yu-Ning Yang, Su Zhang. On linear convergence of projected gradient method for a class of affine rank minimization problems. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1507-1519. doi: 10.3934/jimo.2016.12.1507

 Impact Factor: 

Metrics

  • PDF downloads (103)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]