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Semilocal Convergence Analysis for MMN-HSS Methods under Hölder Conditions

Part of: Numerical linear algebra Nonlinear algebraic or transcendental equations

Published online by Cambridge University Press:  02 May 2017

Yang Li  and
Xue-Ping Guo
Yang Li*
Affiliation:
Department of Mathematics, East China Normal University, Shanghai 200241, China
Xue-Ping Guo*
Affiliation:
Department of Mathematics, East China Normal University, Shanghai 200241, China
*
*Corresponding author. Email addresses:[email protected] (Y. Li), [email protected] (X.-P. Guo)
*Corresponding author. Email addresses:[email protected] (Y. Li), [email protected] (X.-P. Guo)
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Abstract

Multi-step modified Newton-HSS (MMN-HSS) methods, which are variants of inexact Newton methods, have been shown to be competitive for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices. Previously, we established these MMN-HSS methods under Lipschitz conditions, and we now present a semilocal convergence theorem assuming the nonlinear operator satisfies milder Hölder continuity conditions. Some numerical examples demonstrate our theoretical analysis.

Keywords

MMN-HSS methodlarge sparse systems of nonlinear equationHölder conditionspositive-definite Jacobian matricessemilocal convergence

MSC classification

Secondary: 65F10: Iterative methods for linear systems 65F50: Sparse matrices 65H10: Systems of equations
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 2 , May 2017 , pp. 396 - 416
Copyright
Copyright © Global-Science Press 2017 

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References

[1] An, H.B. and Bai, Z.Z., A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations, Appl. Numer. Math. 57, 235–252 (2007).Google Scholar
[2] An, H.B., Mo, Z.Y. and Liu, X.P., A choice of forcing terms in inexact Newton method, J. Comput. Appl. Math. 200, 47–60 (2007).Google Scholar
[3] Bai, Z.Z., A class of two-stage iterative methods for systems of weakly nonlinear equations, Numer. Algor. 14, 295–319 (1997).Google Scholar
[4] Bai, Z.Z., Benzi, M., Chen, F. and Wang, Z.-Q., Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems, IMA J. Numer. Anal. 33, 343–369 (2013).Google Scholar
[5] Bai, Z.Z. and Golub, G.H., Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal. 27, 1–23 (2007).Google Scholar
[6] Bai, Z.Z. and Guo, X.P., On Newton-HSS methods for systems of nonlinear equations with positive-definite Jacobian matrices, J. Comput. Math. 28, 235–260 (2010).Google Scholar
[7] Bai, Z.Z., Golub, G.H. and Li, C.K., Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices, SIAM J. Sci. Comput. 28, 583–603 (2006).Google Scholar
[8] Bai, Z.Z., Golub, G.H. and Li, C.K., Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices, Math. Comput. 76, 287–298 (2007).Google Scholar
[9] Bai, Z.Z., Golub, G.H., Lu, L.Z. and Yin, J.F., Block triangular and skew-Hermitian splitting methods for positive-definite linear systems, SIAM J. Sci. Comput. 26, 844–863 (2005).CrossRefGoogle Scholar
[10] Bai, Z.Z., Golub, G.H. and Ng, M.K., Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl. 24, 603–626 (2003).Google Scholar
[11] Bai, Z.Z., Golub, G.H. and Pan, J.Y., Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math. 98, 1–32 (2004).Google Scholar
[12] Benzi, M., Gander, M.J. and Golub, G.H., Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems, BIT Numer. Math. 43, 881–900 (2003).Google Scholar
[13] Brown, P.N. and Saad, Y., Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Sci. Sta. Comput. 11, 450–481 (1990).Google Scholar
[14] Brown, P.N. and Saad, Y., Convergence theory of nonlinear Newton-Krylov algorithms, SIAM J. Opt. 4, 297–330 (1994).Google Scholar
[15] Chen, M.H., Lin, R.F. and Wu, Q.B., Convergence analysis of the modified Newton-HSS method under the Hölder continuity condition, J. Comput. Appl. Math. 264, 115–130 (2014).Google Scholar
[16] Darvishi, M.T. and Barati, A., A third-order Newton-type method to solve systems of nonlinear equations, Appl. Math. Comput. 187, 630–635 (2007).Google Scholar
[17] Dembo, R.S., Eisenstat, S.C. and Steihaug, T., Inexact Newton methods, SIAM J. Numer. Anal. 19, 400–408 (1982).Google Scholar
[18] Eisenstat, S.C. and Walker, H.F., Globally convergent inexact Newton methods, SIAM J. Opt. 4, 393–422 (1994).Google Scholar
[19] Eisenstat, S.C. and Walker, H.F., Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput. 17, 16–32 (1996).Google Scholar
[20] Elman, H., Silvester, D. and Wathen, A., Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics, Oxford University Press: UK (2014).Google Scholar
[21] Guo, X.P., On the convergence of Newton's method in Banach space, J. Zhejiang Univ. (Sci. Edit.) 27, 484–492 (2000).Google Scholar
[22] Guo, X.P., On semilocal convergence of inexact Newton methods, J. Comput. Math. 25, 231–242 (2007).Google Scholar
[23] Guo, X.P. and Duff, I.S., Semilocal and global convergence of the Newton-HSS method for systems of nonlinear equations, Numer. Linear Algebra Appl. 18, 299–315 (2011).CrossRefGoogle Scholar
[24] Kantorovich, L.V. and Akilov, G.P., Functional Analysis, Pergamon Press: Oxford (1982).Google Scholar
[25] Kelley, C.T., Iterative Methods for Linear and Nonlinear Equations, SIAM: Philadelphia (1995).Google Scholar
[26] Li, Y. and Guo, X.P., Multi-step modified Newton-HSS method for systems of nonlinear equations with positive definite Jacobian matrices, Numer. Algor., DOI: 10.1007/s11075-016-0196-6 (2016).Google Scholar
[27] Ortega, J.M. and Rheinboldt, W.C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press: New York (1970).Google Scholar
[28] Pernice, M. and Walker, H.F., NITSOL: A Newton iterative solver for nonlinear systems, SIAM J. Sci. Comput. 19, 302–318 (1998).CrossRefGoogle Scholar
[29] Rheinboldt, W.C., Methods of Solving Systems of Nonlinear Equations, 2nd Ed. SIAM: Philadelphia (1998).Google Scholar
[30] Wu, Q.B. and Chen, M.H., Convergence analysis of modified Newton-HSS method for solving systems of nonlinear equations, Numer. Algor. 64, 659–683 (2013).Google Scholar
[31] Saad, Y., Iterative Methods for Sparse Linear Systems, 2nd Ed., SIAM: Philadelphia (2003).Google Scholar
[32] Shen, W.P. and Li, C., Convergence criterion of inexact methods for operators with Hölder continuity derivatives, Taiwanese J. Math. 12, 1865–1882 (2008).CrossRefGoogle Scholar
[33] Wang, X.H. and Guo, X.P., On the unified determination for the convergence of Newton's method and its deformations, Numer. Math. J. Chinese Univ. 4, 363–368 (1999).Google Scholar