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

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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.

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References

  1. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    MATH  Google Scholar 

  2. Rheinboldt, W.C.: Methods of Solving Systems of Nonlinear Equations, The 2nd Edn. SIAM, Philadelphia (1998)

    Book  Google Scholar 

  3. Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton mehtods. SIAM J. Numer. Anal. 19, 400–408 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  4. Guo, X.-P.: On semilocal convergence of inexact Newton methods. J. Comput. Math. 25, 231–242 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Eisenstat, S.C., Walker, H.F.: Globally convergent inexact Newton methods. SIAM J. Optim. 4, 393–422 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia (1995)

    Book  MATH  Google Scholar 

  7. Saad, Y.: Iterative Methods for Sparse Linear Systems, The 2nd edn. SIAM, Philadelphia (2003)

    Book  Google Scholar 

  8. Brown, P.N., Saad, Y.: Convergence theory of nonlinear Newton–Krylov algorithms. SIAM J. Optim. 4, 297–330 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brown, P.N., Saad, Y.: Hybrid Krylov methods for nonlinear systems of equations. SIAM J. Sci. Statist. Comput. 11, 450–481 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Eisenstat, S.C., Walker, H.F.: Globally convergent inexact Newton methods. SIAM J. Optim. 4, 393–422 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Bargiacchi-Soula, S., Fehrenbach, J., Masmoudi, M.: From linear to nonlinear large scale systems. SIAM J. Matrix. Anal. Appl. 31, 1552–1569 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, Zhejiang, People’s Republic of China

    Qingbiao Wu & Minhong Chen

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  1. Qingbiao Wu
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  2. Minhong Chen
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Correspondence to Minhong Chen.

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Wu, Q., Chen, M. Convergence analysis of modified Newton-HSS method for solving systems of nonlinear equations. Numer Algor 64, 659–683 (2013). https://doi.org/10.1007/s11075-012-9684-5

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  • DOI: https://doi.org/10.1007/s11075-012-9684-5

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