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Modified Newton-AGSOR method for solving nonlinear systems with block two-by-two complex symmetric Jacobian matrices

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Abstract

In this paper, we modify the accelerated generalized successive overrelaxation (AGSOR) method for block two-by-two complex linear systems, and use the AGSOR method as an inner iteration for the modified Newton equations to solve the nonlinear system whose Jacobian matrix is a block two-by-two complex symmetric matrix. Our new method is named modified Newton AGSOR (MN-AGSOR) method. Because generalized successive overrelaxation (GSOR) method is a special form of the AGSOR method, the modified Newton GSOR (MN-GSOR) method is also analyzed in the discussion. Next, we use the Hölder continuous condition instead of the Lipschitz assumption to analyze and prove the local convergence properties of the MN-AGSOR method. At last, numerical experiments verify the efficiency of the MN-AGSOR method, and it can be seen from the comparison of various aspects that the MN-AGSOR method is superior to some other recently proposed methods.

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References

  1. Bai, Z.Z.: Block preconditioners for elliptic PDE-constrained optimization problems. Computing 91(4), 379–395 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bai, Z.Z., Benzi, M., Chen, F.: Modified HSS iteration methods for a class of complex symmetric linear systems. Computing 87(3–4), 93–111 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bai, Z.Z., Benzi, M., Chen, F.: On preconditioned MHSS iteration methods for complex symmetric linear systems. Numer. Algorithms 56(2), 297–317 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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(3), 603–626 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, M., Lin, R., Wu, Q.: Convergence analysis of the modified Newton-HSS method under the Hölder continuous condition. J. Comput. Appl. Math. 264, 115–130 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, M.H., Wu, Q.B.: Modified Newton-MDPMHSS method for solving nonlinear systems with block two-by-two complex symmetric Jacobian matrices. Numer. Algorithms 80(2), 355–375 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  7. Darvishi, M., Barati, A.: A third-order Newton-type method to solve systems of nonlinear equations. Appl. Math. Comput. 187(2), 630–635 (2007)

    MathSciNet  MATH  Google Scholar 

  8. Dehghan, M., Dehghani-Madiseh, M., Hajarian, M.: A generalized preconditioned MHSS method for a class of complex symmetric linear systems. Math. Model. Anal. 18(4), 561–576 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Edalatpour, V., Hezari, D., Khojasteh Salkuyeh, D.: Accelerated generalized SOR method for a class of complex systems of linear equations. Math. Commun. 20(1), 37–52 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Edalatpour, V., Hezari, D., Salkuyeh, D.: Two efficient inexact algorithms for a class of large sparse complex linear systems. Mediterr. J. Math. 13(4), 2301–2318 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hezari, D., Edalatpour, V., Salkuyeh, D.: Preconditioned GSOR iterative method for a class of complex symmetric system of linear equations. Numer. Linear Algebra Appl. 22(4), 761–776 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Karlsson, H.O.: The quasi-minimal residual algorithm applied to complex symmetric linear systems in quantum reactive scattering. J. Chem. Phys. 103(12), 4914–4919 (1995)

    Article  Google Scholar 

  13. Li, C.X., Wu, S.L.: A single-step HSS method for non-Hermitian positive definite linear systems. Appl. Math. Lett. 44, 26–29 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lions, J.L.: Optimal control of systems governed by partial differential equations problèmes aux limites. Springer, Berlin (1971)

    Book  Google Scholar 

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

    MATH  Google Scholar 

  16. Papp, D., Vizvari, B.: Effective solution of linear Diophantine equation systems with an application in Chemistry. J. Math. Chem. 39(1), 15–31 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rees, T., Dollar, H.S., Wathen, A.J.: Optimal solvers for PDE-constrained optimization. SIAM J. Sci. Comput. 32(1), 271–298 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Salkuyeh, D.K., Hezari, D., Edalatpour, V.: Generalized successive overrelaxation iterative method for a class of complex symmetric linear system of equations. Int. J. Comput. Math. 92(4), 802–815 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, J., Guo, X.P., Zhong, H.X.: MN-DPMHSS iteration method for systems of nonlinear equations with block two-by-two complex Jacobian matrices. Numer. Algorithms 77(1), 167–184 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang, X., Xiao, X.Y., Zheng, Q.Q.: A single-step iteration method for non-Hermitian positive definite linear systems. J. Comput. Appl. Math. 346, 471–482 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wu, Q., Chen, M.: Convergence analysis of modified Newton-HSS method for solving systems of nonlinear equations. Numer. Algorithms 64(4), 659–683 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Xiao, X.Y., Wang, X., Yin, H.W.: Efficient single-step preconditioned HSS iteration methods for complex symmetric linear systems. Comput. Math. Appl. 74(10), 2269–2280 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, School of Science, Nanchang University, 999 Xuefu Road, Nanchang, 330031, People’s Republic of China

    Xin Qi, Hui-Ting Wu & Xiao-Yong Xiao

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  1. Xin Qi
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  2. Hui-Ting Wu
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  3. Xiao-Yong Xiao
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Correspondence to Xin Qi or Xiao-Yong Xiao.

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Qi, X., Wu, HT. & Xiao, XY. Modified Newton-AGSOR method for solving nonlinear systems with block two-by-two complex symmetric Jacobian matrices. Calcolo 57, 14 (2020). https://doi.org/10.1007/s10092-020-00362-w

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