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

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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
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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.
[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.
[3]	Z. Z. Bai, On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems,, Computing, 89 (2010), 171.
[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.
[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.
[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.
[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.
[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.
[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.
[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. doi: 10.1137/0719025.
[12]	P. Deuflhard, "Newton Methods for Nonlinear Problems,", Springer-Verlag, (2004).
[13]	S. C. Eisenstat and H. F. Walker, Globally convergent inexact Newton methods,, SIAM J. Optim., 4 (1994), 393. 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. 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. doi: 10.1002/nla.713.
[16]	C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations,", SIAM, (1995). doi: 10.1137/1.9781611970944.
[17]	Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence,", Dover Publications, (2003).
[18]	J. M. Ortega and W. C. Rheinboldt, "Iterative Solution of Nonlinear Equations in Several Variables,", SIAM, (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. doi: 10.1137/S1064827596303843.
[20]	Y. Saad, "Iterative Methods for Sparse Linear Systems,", 2^nd edition, (2003). doi: 10.1137/1.9780898718003.
[21]	C. Sulem and P. L. Sulem, "The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse,", Springer Verlag, (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. doi: 10.1016/j.amc.2009.12.032.

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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.
[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.
[3]	Z. Z. Bai, On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems,, Computing, 89 (2010), 171.
[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.
[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.
[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.
[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.
[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.
[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.
[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. doi: 10.1137/0719025.
[12]	P. Deuflhard, "Newton Methods for Nonlinear Problems,", Springer-Verlag, (2004).
[13]	S. C. Eisenstat and H. F. Walker, Globally convergent inexact Newton methods,, SIAM J. Optim., 4 (1994), 393. 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. 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. doi: 10.1002/nla.713.
[16]	C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations,", SIAM, (1995). doi: 10.1137/1.9781611970944.
[17]	Y. Kuramoto, "Chemical Oscillations, Waves, and Turbulence,", Dover Publications, (2003).
[18]	J. M. Ortega and W. C. Rheinboldt, "Iterative Solution of Nonlinear Equations in Several Variables,", SIAM, (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. doi: 10.1137/S1064827596303843.
[20]	Y. Saad, "Iterative Methods for Sparse Linear Systems,", 2^nd edition, (2003). doi: 10.1137/1.9780898718003.
[21]	C. Sulem and P. L. Sulem, "The Nonlinear Schrödinger Equation, Self-focusing and Wave Collapse,", Springer Verlag, (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. doi: 10.1016/j.amc.2009.12.032.

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