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Increasing the order of convergence for iterative methods to solve nonlinear systems

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Abstract

For solving nonlinear systems, we introduce a technique that improves the order of convergence of any given iterative method which uses the extended Newton iteration as a predictor. Based on a given iterative method of order \(p\ge 2\), a new method of order \(p+2\) is developed by introducing just one evaluation of the function. We obtain some new methods with higher order of convergence by applying this procedure to some known methods. Computational efficiency in the general form is discussed and comparisons are made between these new methods and the ones from which have been derived. We also perform several numerical tests to show the asymptotic behaviors which confirm the theoretical results.

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References

  1. Amat, S., Busquier, S., Gutiérrez, J.M.: Geometrical constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157, 197–205 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: Increasing the convergence order of an iterative method for nonlinear systems. Appl. Math. Lett. 25, 2369–2374 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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 

  4. Darvishi, M.T., Barati, A.: A fourth-order method from quadrature formulae to solve systems of nonlinear equations. Appl. Math. Comput. 188, 257–261 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., Zimmermann, P.: MPFR: a multiple-precision binary floating-point library with correct rounding. ACM Trans. Math. Software 33(2), 15. Art. 13 (2007)

    Article  MathSciNet  Google Scholar 

  6. Frontini, M., Sormani, E.: Some variant of Newton’s method with third-order convergence. Appl. Math. Comput. 140, 419–426 (2003)

    MathSciNet  MATH  Google Scholar 

  7. 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 

  8. Grau-Sánchez, M., Grau, A., Noguera, M.: Frozen divided difference scheme for solving systems of nonlinear equations. J. Comput. Appl. Math. 235, 1739–1743 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Grau-Sánchez, M., Grau, A., Noguera, M.: On the computational efficiency index and some iterative methods for solving systems of nonlinear equations. J. Comput. Appl. Math. 236, 1259–1266 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grau-Sánchez, M., Grau, A., Noguera, M.: Ostrowski type methods for solving systems of nonlinear equations. Appl. Math. Comput. 218, 2377–2385 (2011)

    MathSciNet  MATH  Google Scholar 

  11. Gutiérrez, J.M., Hernández, M.A.: A family of Chebyshev-Halley type methods in Banach spaces. Bull. Aust. Math. Soc. 55, 113–130 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Homeier, H.H.H.: A modified Newton method with cubic convergence: the multivariable case. J. Comput. Appl. Math. 169, 161–169 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Homeier, H.H.H.: On Newton-type methods with cubic convergence. J. Comput. Appl. Math. 176, 425–432 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kelley, C.T.: Solving nonlinear equations with Newton’s method. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  15. Kou, J., Li, Y., Wang, X.: Some modification of Newton’s method with fifth-order convergence. J. Comput. Appl. Math. 209, 146–152 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Noor, M.A., Waseem, M.: Some iterative methods for solving a system of nonlinear equations. Comput. Math. Appl. 57, 101–106 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ortega, J.M., Rheinboldt, W.C.: Iterative solutions of nonlinear equations in several variables. Academic Press, New York (1970)

    MATH  Google Scholar 

  18. Ostrowski, A.M.: Solution of equations and systems of equations. Academic Press, New York (1966)

    MATH  Google Scholar 

  19. Palacios, M.: Kepler equation and accelerated Newton method. J. Comput. Appl. Math. 138, 335–346 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sharma, J.R., Arora, H.: Efficient Jarratt-like methods for solving systems of nonlinear equations. Calcolo 51, 193–210 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sharma, J.R., Guha, R.K., Sharma, R.: An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numer. Algorithms 62, 307–323 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sharma, J.R., Gupta, P.: An efficient fifth order method for solving systems of nonlinear equations. Comput. Math. Appl. 67, 591–601 (2014)

    Article  MathSciNet  Google Scholar 

  23. Xiao, X.Y., Yin, H.W.: A new class of methods with higher order of convergence for solving systems of nonlinear equations. Appl. Math. Comput. 264, 300–309 (2015)

    MathSciNet  Google Scholar 

  24. Xiao, X.Y., Yin, H.W.: A simple and efficient method with high order convergence for solving systems of nonlinear equations. Comput. Math. Appl. 69, 1220–1231 (2015)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

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

    X. Y. Xiao & H. W. Yin

Authors
  1. X. Y. Xiao
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  2. H. W. Yin
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Correspondence to X. Y. Xiao.

Additional information

This work was supported by the Natural Science Foundation of Jiangxi Province of China (Grant No. 20151BAB201021) and the National Natural Science Foundation of China (Grant No. 11401293).

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Xiao, X.Y., Yin, H.W. Increasing the order of convergence for iterative methods to solve nonlinear systems. Calcolo 53, 285–300 (2016). https://doi.org/10.1007/s10092-015-0149-9

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  • DOI: https://doi.org/10.1007/s10092-015-0149-9

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