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A new iterative method for finding the multiple roots of nonlinear equations

  • Reza DehghanEmail author
Article
  • 33 Downloads

Abstract

In this paper, an iterative method for multiple roots of nonlinear equations is presented. Its convergence order is analyzed and proved. It is shown that the proposed method has third-order convergence. To assess the validity and performance of the proposed method, some nonlinear equations are solved and the results are compared with the results from six other methods.

Keywords

Nonlinear equation Newton’s method Multiple roots Iterative method 

Mathematics Subject Classification

34Axx 49M15 65Hxx 65F08 

Notes

References

  1. 1.
    Li, S.G., Cheng, L.Z., Neta, B.: Some fourth-order nonlinear solvers with closed formulae for multiple roots. Computers and Mathematics with Applications 59, 126–135 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Hansanov, V. I., Ivanov, I. G., Nedjibov, G.: A new modification of newton’s method, Application of Mathematics in Engineering’27, Proc. of the XXVII Summer School Sozopol’01 278–286 (2002)Google Scholar
  3. 3.
    Cordero, A., Torregrosa, J.R.: Variants of newton’s method using fifth-order quadrature formulas. Applied Mathematics and Computation 190(1), 686–698 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Noor, M.A., Noor, K.I., Al-Said, E., Waseem, M.: Some new iterative methods for nonlinear equations. Mathematical Problems in Engineering (2010).  https://doi.org/10.1155/2010/198943 MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ö zban, A.Y.: Some new variants of newton’s method. Applied Mathematics Letters 17(6), 677–682 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Choubey, N., Jaiswal, J.P.: A derivative-free method of eighth-order for finding simple root of nonlinear equations. Communications in Numerical Analysis 2015(2), 90–103 (2015).  https://doi.org/10.5899/2015/cna-00227 MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fernandez-Torres, G.: A novel geometric modification to the newton-secant method to achieve convergence of order 1+2 and its dynamics, Modelling and Simulation in Engineering 2015, 6 pages (2015)  https://doi.org/10.1155/2015/502854
  8. 8.
    Mahmoodpur, F., Haghighi, A.H.: A new correction of newton’s proce dure to achieve convergence of higher order. Communications on Advanced Computational Science with Applications 2015(2), 65–71 (2015).  https://doi.org/10.5899/2015/cacsa-00045 CrossRefGoogle Scholar
  9. 9.
    Sharma, J.R., Guha, R.K., Sharma, R.: A unified approach to generate weighted newton third order methods for solving nonlinear equations. 105 Journal of Numerical Mathematics and Stochastics 4(1), 48–58 (2012)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chen, J.Y., Kincaid, D.R., Lin, B.R.: A Variant of Newton’s Method Based on Simpson’s Three-eighths Rule for Nonlinear Equations. Applied Mathematics Letters (2017).  https://doi.org/10.1016/j.aml.2017.11.014 Google Scholar
  11. 11.
    Li, X., Mu, C., Ma, J., Hou, L.: Fifth-order iterative method for finding multiple roots of nonlinear equations. Numer Algor 57, 389–398 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Halley, E.: A new, exact and easy method of finding the roots of equations generally and that without any previous reduction. Phil. Trans. R. Soc. London 18, 136–148 (1694)CrossRefGoogle Scholar
  13. 13.
    King, R.F.: A family of fourth order methods for nonlinear equations. SIAM J. Numer. Anal. 10, 876–879 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dong, C.: A family of multipoint iterative functions for finding multiple roots of equations. Int. J. Comput. Math. 21, 363–367 (1987)CrossRefzbMATHGoogle Scholar
  15. 15.
    Zhoua, X., Chena, X., Songa, Y.: Constructing higher-order methods for obtaining the multiple roots of nonlinear equations. Journal of Computational and Applied Mathematics 235, 4199–4206 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jarratt, P.: Some fourth order multipoint methods for solving equations. Math. Comp. 20, 434–437 (1966)CrossRefzbMATHGoogle Scholar
  17. 17.
    Sharma, J.R., Sharma, R.: Modified Jarratt method for computing multiple roots. Appl. Math. Comput. 217, 878–881 (2010)MathSciNetzbMATHGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Masjed-Soleiman BranchIslamic Azad University (I.A.U)Masjed-SoleimanIran

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