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A Family of Optimal Eighth Order Multiple Root Finders with Multivariate Weight Function

  • Fiza ZafarEmail author
  • Alicia Cordero
  • Juan Ramon Torregrosa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11386)

Abstract

Finding repeated zero for a nonlinear equation \(f(x)=0\), \(f:I\subseteq R\rightarrow R\), has always been of much interest and attention due to it’s wide applications in many fields of science and engineering. The modified Newton’s method is usually applied to solve this problem. Keeping in view that very few optimal higher order convergent methods exist for multiple roots, we present a new family of optimal eighth order convergent iterative methods for multiple roots with known multiplicity involving multivariate weight function. The numerical performance of the proposed methods is analyzed extensively along with the basins of attractions. Real life models from Life Science, Engineering and Physics are considered for the sake of comparison. The numerical experiments show that our proposed methods are efficient for determining multiple roots of non-linear equations.

Keywords

Nonlinear equations Multiple zeros Optimal iterative methods Higher order of convergence 

Notes

Acknowledgments

This work is supported by Schlumberger Foundation-Faculty for Future Program, by Ministerio de Economía y Competitividad under grants MTM 2014-52016-C2-2-P and Generalitat Valenciana PROMETEO/2016/089.

References

  1. 1.
    Schroder, E.: Uber unendlich viele algorithmen zur auflosung der gleichungen. Math. Ann. 2, 317–365 (1870)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach. 21, 643–651 (1974)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Geum, Y.H., Kim, Y.I., Neta, B.: A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics. Appl. Math. Comput. 270, 387–400 (2015)MathSciNetGoogle Scholar
  4. 4.
    Geum, Y.H., Kim, Y.I., Neta, B.: A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points. Appl. Math. Comput. 283, 120–140 (2016)MathSciNetGoogle Scholar
  5. 5.
    Behl, R., Cordero, A., Motsa, S.S., Torregrosa, J.R.: Aneighth-order family of optimal multiple root finders and its dynamics. Numer. Algor. 77(4), 1249–1272 (2017).  https://doi.org/10.1007/s11075-017-0361-6CrossRefzbMATHGoogle Scholar
  6. 6.
    Zafar, F., Cordero, A., Rana, Q., Torregrosa, J.R.: Optimal iterative methods for finding multiple roots of nonlinear equations using free parameters. J. Math. Chem. 56(7), 1884–1901 (2018).  https://doi.org/10.1007/s10910-017-0813-1MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Behl, R., Zafar, F., Alshomrani, A.S., Junjua, M., Yasmin, N.: An optimal eighth-order scheme for multiple zeros of univariate function. Int. J. Comput. Math. 15(3), 14 (2018).  https://doi.org/10.1142/S0219876218430028CrossRefGoogle Scholar
  8. 8.
    Behl, R., Alshomrani, A.S., Motsa, S.S.: An optimal schemefor multiple roots of nonlinear equations with eighth-order convergence. J. Math. Chem. 56(7), 1–6 (2018).  https://doi.org/10.1007/s10910-018-0857-xMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jay, L.O.: A note on Q-order of convergence. BIT Numer. Math. 41, 422–429 (2001)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Edelstein-Keshet, L.: Differential Calculus for the Life Sciences. Univeristy of British Columbia, Vancouver (2017)Google Scholar
  11. 11.
    Zachary, J.L.: Introduction to Scientific Programming: Computational Problem Solving Using Maple and C. Springer, New York (2012)Google Scholar
  12. 12.
    Khoury, R., Harder, D.H.: Numerical Methods and Modelling for Engineering. Springer, Switzerland (2017)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Fiza Zafar
    • 1
    • 2
    Email author
  • Alicia Cordero
    • 1
  • Juan Ramon Torregrosa
    • 1
  1. 1.Instituto de Matemática MultidisciplinarUniversitat Politècnica de ValènciaValènciaSpain
  2. 2.Centre for Advanced Studies in Pure and Applied MathematicsBahauddin Zakariya UniversityMultanPakistan

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