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The Derivative-Free Double Newton Step Methods for Solving System of Nonlinear Equations

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Abstract

In this paper, we propose two classes of derivative-free Newton-like methods for solving system of nonlinear equations based on double Newton step. We also give the local convergence analysis of the iterative methods. In addition, some numerical results are also reported in the paper, which confirm the good theoretical properties of our approach.

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References

  1. Bai Z.Z.: A class of two-stage iterative methods for systems of weakly nonlinear equations. Numer. Algorithms 14, 295–319 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bai Z.Z., Yang X.: On HSS-based iteration methods for weakly nonlinear systems. Appl. Numer. Math. 59, 2923–2936 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Huang N., Ma C.F.: A regularized smoothing Newton method for solving SOCCPs based on a new smoothing C-function. Appl. Math. Comput. 230, 315–329 (2014)

    MathSciNet  Google Scholar 

  4. Narushima Y., Sagara N., Ogasawara H.: A smoothing Newton method with Fischer–Burmeister function for second-order cone complementarity problems. J. Optim. Theory Appl. 149, 79–101 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Darvishi M.T., Barati A.: Super cubic iterative methods to solve systems of nonlinear equations. Appl. Math. Comput. 188, 1678–1685 (2007)

    MathSciNet  MATH  Google Scholar 

  6. Huang, N.; Ma, C.F.: Convergence analysis and numerical study of a fixed-point iterative method for solving systems of nonlinear equations. Sci. World J. 2014 Article ID 789459, 10

  7. Cordero A., Torregrosa J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)

    MathSciNet  MATH  Google Scholar 

  8. Frontini M., Sormani E.: Third-order methods from quadrature formulae for solving systems of nonlinear equations. Appl. Math. Comput. 149, 771–782 (2004)

    MathSciNet  MATH  Google Scholar 

  9. Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Three-step iterative methods with optimal eighth-order convergence. J. Comput. Appl. Math. 235, 3189–3194 (2011)

  10. Grau M., Noguera M.: A variant of Cauchys method with accelerated fifth-order convergence. Appl. Math. Lett. 17, 509–517 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grau-Snchez, M., Noguera, M., Amat, S.: On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods. J. Comput. Appl. Math. 237, 363–372 (2013)

  12. Argyros I.K., Hilout S.: On the local convergence of fast two-step Newton-like methods for solving nonlinear equations. J. Comput. Appl. Math. 245, 1–9 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ezquerro J.A., Hernndez M.A.: An optimization of Chebyshevs method. J. Complex. 25, 343–361 (2009)

    Article  MathSciNet  Google Scholar 

  14. Argyros I.K., Ezquerro J.A., Gutirrez J.M., Hernndez M.A., Hilout S.: On the semilocal convergence of efficient Chebyshev–Secant-type methods. J. Comput. Appl. Math. 235, 3195–3206 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Argyros I.K.: Convergence and Applications of Newton-Type Iterations. Springer-Verlag Publ., New York (2008)

    MATH  Google Scholar 

  16. Kantorovich L.V., Akilov G.P.: Functional Analysis. Pergamon Press, Oxford (1982)

    MATH  Google Scholar 

  17. Argyros I.K., Cho Y.J., Hilout S.: Numerical Methods for Equations and Its Applications. CRC Press/Taylor and Francis Group, NY (2012)

    MATH  Google Scholar 

  18. Chandrasekhar S.: Radiative Transfer. Dover Publ., New York (1960)

    MATH  Google Scholar 

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

  20. Barbero A.C., Molada E.M., Sánchez J.R.T.: Iterative methods of order four and five for systems of nonlinear equations. J. Comput. Appl. Math. 231, 541–551 (2009)

    Article  MathSciNet  Google Scholar 

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Correspondence to Changfeng Ma.

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Huang, N., Ma, C. & **e, Y. The Derivative-Free Double Newton Step Methods for Solving System of Nonlinear Equations. Mediterr. J. Math. 13, 2253–2270 (2016). https://doi.org/10.1007/s00009-015-0581-8

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  • DOI: https://doi.org/10.1007/s00009-015-0581-8

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