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A study of the local convergence of a fifth order iterative method

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Abstract

We present a local convergence study of a fifth order iterative method to approximate a locally unique root of nonlinear equations. The analysis is discussed under the assumption that first order Fréchet derivative satisfies the Lipschitz continuity condition. Moreover, we consider the derivative free method that obtained through approximating the derivative with divided difference along with the local convergence study. Finally, we provide computable radii and error bounds based on the Lipschitz constant for both cases. Some of the numerical examples are worked out and compared the results with existing methods.

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References

  1. A. Constantinides and N. Mostoufi, Numerical Methods for Chemical Engineers with MATLAB Applications, Prentice Hall PTR, New Jersey, (1999).

    Google Scholar 

  2. J. M. Douglas, Process Dynamics and Control, Prentice Hall, Englewood Cliffs, (1972).

    Google Scholar 

  3. M. Shacham, An improved memory method for the solution of a nonlinear equation, Chem. Eng. Sci., 44 (1989), 1495–1501.

    Article  Google Scholar 

  4. J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New-York, (1970).

    MATH  Google Scholar 

  5. J. R. Sharma and H. Arora, A novel derivative free algorithm with seventh order convergence for solving systems of nonlinear equations, Numer. Algorithms, 67 (2014), 917–933.

    Article  MathSciNet  Google Scholar 

  6. I. K. Argyros, A. A. Magreńan, and L. Orcos, Local convergence and a chemical application of derivative free root finding methods with one parameter based on interpolation, J. Math. Chem., 54 (2016), 1404–1416.

    Article  MathSciNet  Google Scholar 

  7. E. L. Allgower and K. Georg, Lectures in Applied Mathematics, American Mathematical Society (Providence, RI) 26, 723–762.

  8. A. V. Rangan, D. Cai, and L. Tao, Numerical methods for solving moment equations in kinetic theory of neuronal network dynamics, J. Comput. Phys., 221 (2007), 781–798.

    Article  MathSciNet  Google Scholar 

  9. A. Nejat and C. Ollivier-Gooch, Effect of discretization order on preconditioning and convergence of a high-order unstructured Newton-GMRES solver for the Euler equations, J. Comput. Phys., 227 (2008), 2366–2386.

    Article  MathSciNet  Google Scholar 

  10. C. Grosan and A. Abraham, A new approach for solving nonlinear equations systems, IEEE Trans. Syst. Man Cybernet Part A: System Humans, 38 (2008), 698–714.

    Article  Google Scholar 

  11. F. Awawdeh, On new iterative method for solving systems of nonlinear equations, Numer. Algorithms, 54 (2010), 395–409.

    Article  MathSciNet  Google Scholar 

  12. I. G. Tsoulos and A. Stavrakoudis, On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods, Nonlinear Anal. Real World Appl., 11 (2010), 2465–2471.

    Article  MathSciNet  Google Scholar 

  13. E. Martínez, S. Singh, J. L. Hueso, and D. K. Gupta, Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces, Appl. Math. Comput., 281 (2016), 252–265.

    MathSciNet  MATH  Google Scholar 

  14. S. Singh, D. K. Gupta, E. Martínez, and J. L. Hueso, Semi local and local convergence of a fifth order iteration with Fréchet derivative satisfying Hölder condition, Appl. Math. Comput., 276 (2016), 266–277.

    MathSciNet  MATH  Google Scholar 

  15. I. K. Argyros and S. George, Local convergence of modified Halley-like methods with less computation of inversion, Novi. Sad.J. Math., 45 (2015), 47–58.

    Article  MathSciNet  Google Scholar 

  16. I. K. Argyros, R. Behl, and S. S. Motsa, Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First Derivative Algorithms 2015, 8, 1076–1087; doi:https://doi.org/10.3390/a8041076.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. I. K. Argyros and A. A. Magreñán, A study on the local convergence and dynamics of Chebyshev- Halley-type methods free from second derivative, Numer. Algorithms71 (2016), 1–23.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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Acknowledgement

This research was partially supported by Ministerio de Economía y Competitividad under grant PGC2018-095896-B-C21-C22.

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Authors and Affiliations

  1. Dr. B.R. Ambedkar Nationl Instutute of Technology, Jalandhar, India

    Sukjith Singh

  2. Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, València, Spain

    Eulalia Martínez

  3. Amrita School of Engineering, Department of Mathematics, Amrita Vishwa Vidhyapeetham, Coimbatore, India

    P. Maroju

  4. Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Ramandeep Behl

Authors
  1. Sukjith Singh
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  2. Eulalia Martínez
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  3. P. Maroju
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  4. Ramandeep Behl
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Corresponding authors

Correspondence to Sukjith Singh, Eulalia Martínez, P. Maroju or Ramandeep Behl.

Additional information

This research was partially supported by Ministerio de Economa y Competitividad under grant MTM2014-52016-C2-1-2-P and by the project of Generalitat Valenciana Prometeo/2016/089.

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Singh, S., Martínez, E., Maroju, P. et al. A study of the local convergence of a fifth order iterative method. Indian J Pure Appl Math 51, 439–455 (2020). https://doi.org/10.1007/s13226-020-0409-5

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  • DOI: https://doi.org/10.1007/s13226-020-0409-5

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