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Convergence analysis of modified Newton-HSS method for solving systems of nonlinear equations

Published: 01 December 2013 Publication History

Abstract

Hermitian and skew-Hermitian splitting(HSS) method has been proved quite successfully in solving large sparse non-Hermitian positive definite systems of linear equations. Recently, by making use of HSS method as inner iteration, Newton-HSS method for solving the systems of nonlinear equations with non-Hermitian positive definite Jacobian matrices has been proposed by Bai and Guo. It has shown that the Newton-HSS method outperforms the Newton-USOR and the Newton-GMRES iteration methods. In this paper, a class of modified Newton-HSS methods for solving large systems of nonlinear equations is discussed. In our method, the modified Newton method with R-order of convergence three at least is used to solve the nonlinear equations, and the HSS method is applied to approximately solve the Newton equations. For this class of inexact Newton methods, local and semilocal convergence theorems are proved under suitable conditions. Moreover, a globally convergent modified Newton-HSS method is introduced and a basic global convergence theorem is proved. Numerical results are given to confirm the effectiveness of our method.

References

[1]
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)
[2]
Rheinboldt, W.C.: Methods of Solving Systems of Nonlinear Equations, The 2nd Edn. SIAM, Philadelphia (1998)
[3]
Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton mehtods. SIAM J. Numer. Anal. 19, 400---408 (1982)
[4]
Guo, X.-P.: On semilocal convergence of inexact Newton methods. J. Comput. Math. 25, 231---242 (2007)
[5]
Eisenstat, S.C., Walker, H.F.: Globally convergent inexact Newton methods. SIAM J. Optim. 4, 393---422 (1994)
[6]
Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia (1995)
[7]
Saad, Y.: Iterative Methods for Sparse Linear Systems, The 2nd edn. SIAM, Philadelphia (2003)
[8]
Brown, P.N., Saad, Y.: Convergence theory of nonlinear Newton---Krylov algorithms. SIAM J. Optim. 4, 297---330 (1994)
[9]
Brown, P.N., Saad, Y.: Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Statist. Comput. 11, 450---481 (1990)
[10]
An, H.-B., Bai, Z.-Z.: A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations. Appl. Numer. Math. 57, 235---252 (2007)
[11]
Bai, Z.-Z., Golub, G.H., 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)
[12]
Bai, Z.-Z., Golub, G.H., Pan, J.-Y.: Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math. 98, 1---32 (2004)
[13]
Bai, Z.-Z., Golub, G.H., 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)
[14]
Bai, Z.-Z., Golub, G.H., Lu, L.-Z., Yin, J.-F.: Block triangular and skew-Hermitian splitting methods for positive-definite linear systems. SIAM J. Sci. Comput. 26, 844---863 (2005)
[15]
Bai, Z.-Z., Golub, G.H., 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)
[16]
Bai, Z.-Z., Golub, G.H.: Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA J. Numer. Anal. 27, 1---23 (2007)
[17]
Benzi, M., Gander, M.J., Golub, G.H.: Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems. BIT Numer. Math. 43, 881---900 (2003)
[18]
Bai, Z.-Z., Benzi, M., Chen, F., Wang, Z.-Q.: Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems. Technical Report 2011-001, Math/CS Department, Emory University (January 2011). To appear in IMA J. Numer. Anal.
[19]
Bai, Z.-Z., Guo, X.-P.: The Newton-HSS methods for systems of nonlinear equations with positive-definite Jacobian matrices. J. Comput. Math. 28, 235---260 (2010)
[20]
Darvishi, M.T., Barati, A.: A third-order Newton-type method to solve systems of nonlinear equations. Appl. Math. Comput. 187, 630---635 (2007)
[21]
Guo, X.-P., 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)
[22]
Eisenstat, S.C., Walker, H.F.: Globally convergent inexact Newton methods. SIAM J. Optim. 4, 393---422 (1994)
[23]
Bargiacchi-Soula, S., Fehrenbach, J., Masmoudi, M.: From linear to nonlinear large scale systems. SIAM J. Matrix. Anal. Appl. 31, 1552---1569 (2010)

Cited By

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  • (2024)Modified Newton-PBS method for solving a class of complex symmetric nonlinear systemsNumerical Algorithms10.1007/s11075-023-01649-z96:1(333-368)Online publication date: 1-May-2024
  • (2020)Modified Newton-AGSOR method for solving nonlinear systems with block two-by-two complex symmetric Jacobian matricesCalcolo: a quarterly on numerical analysis and theory of computation10.1007/s10092-020-00362-w57:2Online publication date: 14-Mar-2020
  • (2019)Modified Newton-DSS method for solving a class of systems of nonlinear equations with complex symmetric Jacobian matricesNumerical Algorithms10.1007/s11075-019-00847-y85:3(951-975)Online publication date: 17-Dec-2019
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Information

Published In

cover image Numerical Algorithms
Numerical Algorithms  Volume 64, Issue 4
December 2013
161 pages

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 December 2013

Author Tags

  1. Convergence analysis
  2. Hermitian and Skew-Hermitian splitting
  3. Large sparse systems
  4. Newton-HSS method
  5. Nonlinear equations
  6. Positive-definite Jacobian matrices

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View all
  • (2024)Modified Newton-PBS method for solving a class of complex symmetric nonlinear systemsNumerical Algorithms10.1007/s11075-023-01649-z96:1(333-368)Online publication date: 1-May-2024
  • (2020)Modified Newton-AGSOR method for solving nonlinear systems with block two-by-two complex symmetric Jacobian matricesCalcolo: a quarterly on numerical analysis and theory of computation10.1007/s10092-020-00362-w57:2Online publication date: 14-Mar-2020
  • (2019)Modified Newton-DSS method for solving a class of systems of nonlinear equations with complex symmetric Jacobian matricesNumerical Algorithms10.1007/s11075-019-00847-y85:3(951-975)Online publication date: 17-Dec-2019
  • (2019)Modified Newton-MDPMHSS method for solving nonlinear systems with block two-by-two complex symmetric Jacobian matricesNumerical Algorithms10.1007/s11075-018-0488-080:2(355-375)Online publication date: 1-Feb-2019
  • (2018)On the accelerated modified Newton-HSS method for systems of nonlinear equationsNumerical Algorithms10.1007/s11075-018-0472-879:4(1049-1073)Online publication date: 1-Dec-2018
  • (2018)MN-DPMHSS iteration method for systems of nonlinear equations with block two-by-two complex Jacobian matricesNumerical Algorithms10.1007/s11075-017-0309-x77:1(167-184)Online publication date: 1-Jan-2018
  • (2018)Modified Newton-NSS method for solving systems of nonlinear equationsNumerical Algorithms10.1007/s11075-017-0301-577:1(1-21)Online publication date: 1-Jan-2018
  • (2017)Multi-step modified Newton-HSS methods for systems of nonlinear equations with positive definite Jacobian matricesNumerical Algorithms10.1007/s11075-016-0196-675:1(55-80)Online publication date: 1-May-2017
  • (2016)Semilocal convergence analysis for the modified Newton-HSS method under the Holder conditionNumerical Algorithms10.1007/s11075-015-0061-z72:3(667-685)Online publication date: 1-Jul-2016
  • (2015)On preconditioned modified Newton-MHSS method for systems of nonlinear equations with complex symmetric jacobian matricesNumerical Algorithms10.1007/s11075-014-9912-269:3(553-567)Online publication date: 1-Jul-2015

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