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Solving nonlinear problems by Ostrowski–Chun type parametric families

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An Erratum to this article was published on 17 February 2015

Abstract

In this paper, by using a generalization of Ostrowski’ and Chun’s methods two bi-parametric families of predictor–corrector iterative schemes, with order of convergence four for solving system of nonlinear equations, are presented. The predictor of the first family is Newton’s method, and the predictor of the second one is Steffensen’s scheme. One of them is extended to the multidimensional case. Some numerical tests are performed to compare proposed methods with existing ones and to confirm the theoretical results. We check the obtained results by solving the molecular interaction problem.

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References

  1. M.S. Petkovic̀, B. Neta, L.D. Petkovic̀, J. Dz̆unic̀, Multipoint Methods for Solving Nonlinear Equations (Academic, New York, 2013)

    Google Scholar 

  2. M. Mahalakshmi, G. Hariharan, K. Kannan, The wavelet methods to linear and nonlinear reaction–diffusion model arising in mathematical chemistry. J. Math. Chem. 51(9), 2361–2385 (2013)

    Article  CAS  Google Scholar 

  3. P.G. Logrado, J.D.M. Vianna, Partitioning technique procedure revisited: Formalism and first application to atomic problems. J. Math. Chem. 22, 107–116 (1997)

    Article  CAS  Google Scholar 

  4. C.G. Jesudason, I. Numerical nonlinear analysis: differential methods and optimization applied to chemical reaction rate determination. J. Math. Chem. 49, 1384–1415 (2011)

    Article  CAS  Google Scholar 

  5. K. Maleknejad, M. Alizadeh, An efficient numerical scheme for solving hammerstein integral equation arisen in chemical phenomenon. Procedia Comput. Sci. 3, 361–364 (2011)

    Article  Google Scholar 

  6. R.C. Rach, J.S. Duan, A.M. Wazwaz, Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. J. Math. Chem. 52, 255–267 (2014)

    Article  CAS  Google Scholar 

  7. J.F. Steffensen, Remarks on iteration. Skand. Aktuar Tidskr. 16, 64–72 (1933)

    Google Scholar 

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

    Google Scholar 

  9. H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration. J. ACM 21, 643–651 (1974)

    Article  Google Scholar 

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

    Article  Google Scholar 

  11. J.R. Sharma, H. Arora, On efficient weighted-Newton methods for solving systems of nonlinear equations. Appl. Math. Comput. 222, 497–506 (2013)

    Article  Google Scholar 

  12. M. Abad, A. Cordero, J.R. Torregrosa, Fourth- and fifth-order methods for solving nonlinear systems of equations: an application to the Global positioning system. Abstr. Appl. Anal.(2013) Article ID:586708. doi:10.1155/2013/586708

  13. F. Soleymani, T. Lotfi, P. Bakhtiari, A multi-step class of iterative methods for nonlinear systems. Optim. Lett. 8, 1001–1015 (2014)

    Article  Google Scholar 

  14. M.T. Darvishi, N. Darvishi, SOR-Steffensen-Newton method to solve systems of nonlinear equations. Appl. Math. 2(2), 21–27 (2012). doi:10.5923/j.am.20120202.05

    Google Scholar 

  15. F. Awawdeh, On new iterative method for solving systems of nonlinear equations. Numer. Algorithms 5(3), 395–409 (2010)

    Article  Google Scholar 

  16. D.K.R. Babajee, A. Cordero, F. Soleymani, J.R. Torregrosa, On a novel fourth-order algorithm for solving systems of nonlinear equations. J. Appl. Math. (2012) Article ID:165452. doi:10.1155/2012/165452

  17. A. Cordero, J.R. Torregrosa, M.P. Vassileva, Pseudocomposition: a technique to design predictor–corrector methods for systems of nonlinear equations. Appl. Math. Comput. 218(23), 1496–1504 (2012)

    Article  Google Scholar 

  18. A. Cordero, J.R. Torregrosa, M.P. Vassileva, Increasing the order of convergence of iterative schemes for solving nonlinear systems. J. Comput. Appl. Math. 252, 86–94 (2013)

    Article  Google Scholar 

  19. A.M. Ostrowski, Solution of Equations and System of Equations (Academic, New York, 1966)

    Google Scholar 

  20. C. Chun, Construction of Newton-like iterative methods for solving nonlinear equations. Numer. Math. 104, 297–315 (2006)

    Article  Google Scholar 

  21. R. King, A family of fourth order methods for nonlinear equations. SIAM J. Numer. Anal. 10, 876–879 (1973)

    Article  Google Scholar 

  22. A. Cordero, J.R. Torregrosa, Low-complexity root-finding iteration functions with no derivatives of any order of convergence. J. Comput. Appl. Math. (2014). doi:10.1016/j.cam.2014.01.024

  23. A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, A modified Newton Jarratts composition. Numer. Algorithms 55, 87–99 (2010)

    Article  Google Scholar 

  24. P. Jarratt, Some fourth order multipoint methods for solving equations. Math. Comput. 20, 434–437 (1966)

    Article  Google Scholar 

  25. A. Cordero, J.R. Torregrosa, Variants of Newtons method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)

    Article  Google Scholar 

  26. Z. Liu, Q. Zheng, P. Zhao, A variant of Steffensens method of fourth-order convergence and its applications. Appl. Math. Comput. 216, 1978–1983 (2010)

    Article  Google Scholar 

  27. A. Cordero, J.R. Torregrosa, A class of Steffensen type methods with optimal order of convergence. Appl. Math. Comput. 217, 7653–7659 (2011)

    Article  Google Scholar 

  28. L.B. Rall, New York, Computational Solution of Nonlinear Operator Equations (Robert E. Krieger Publishing Company Inc, New York, 1969)

    Google Scholar 

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Authors and Affiliations

  1. Instituto de Matemática Multidisciplinar, Universitat Politécnica de Valencia, Valencia, Spain

    Alicia Cordero & Juan R. Torregrosa

  2. Instituto Tecnológico de Santo Domingo (INTEC), av. Los Próceres, Galá, Santo Domingo, República Dominicana

    Javier G. Maimó & María P. Vassileva

Authors
  1. Alicia Cordero
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  2. Javier G. Maimó
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  3. Juan R. Torregrosa
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  4. María P. Vassileva
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Corresponding author

Correspondence to María P. Vassileva.

Additional information

This research was supported by Ministerio de Ciencia y Tecnología MTM2011-28636-C02-02 and FONDOCYT, República Dominicana.

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Cordero, A., Maimó, J.G., Torregrosa, J.R. et al. Solving nonlinear problems by Ostrowski–Chun type parametric families. J Math Chem 53, 430–449 (2015). https://doi.org/10.1007/s10910-014-0432-z

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  • DOI: https://doi.org/10.1007/s10910-014-0432-z

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