American Institute of Mathematical Sciences

Advanced
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

Ai-Li Yang 1, and  Yu-Jiang Wu 1,

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.
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,", 2nd 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,", 2nd 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

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