Advertisement

Numerical Algorithms

, Volume 77, Issue 4, pp 1249–1272 | Cite as

An eighth-order family of optimal multiple root finders and its dynamics

  • Ramandeep Behl
  • Alicia Cordero
  • Sandile S. Motsa
  • Juan R. Torregrosa
Original Paper

Abstract

There is a very small number of higher-order iteration functions for multiple zeros whose order of convergence is greater than four. Some scholars have tried to propose optimal eighth-order methods for multiple zeros. But, unfortunately, they did not get success in this direction and attained only sixth-order convergence. So, as far as we know, there is not a single optimal eighth-order iteration function in the available literature that will work for multiple zeros. Motivated and inspired by this fact, we present an optimal eighth-order iteration function for multiple zeros. An extensive convergence study is discussed in order to demonstrate the optimal eighth-order convergence of the proposed scheme. In addition, we also demonstrate the applicability of our proposed scheme on real-life problems and illustrate that the proposed methods are more efficient among the available multiple root finding techniques. Finally, dynamical study of the proposed schemes also confirms the theoretical results.

Keywords

Nonlinear equations Optimal iterative methods Multiple roots Efficiency index Kung-Traub conjecture 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

The authors thank to the anonymous referees for their useful comments and suggestions to improve the final version of the manuscript.

References

  1. 1.
    Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach. 21, 643–651 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ostrowski, A.M.: Solution of equations and systems of equations. Academic Press, New York (1960)zbMATHGoogle Scholar
  3. 3.
    Traub, J.F.: Iterative Methods for the solution of equations. Prentice-Hall, Englewood Cliffs (1964)zbMATHGoogle Scholar
  4. 4.
    Petković, M. S., Neta, B., Petković, L. D., Dzunić, J.: Multipoint methods for solving nonlinear equations. Academic Press (2013)Google Scholar
  5. 5.
    Behl, R., Cordero, A., Motsa, S.S., Torregrosa, J.R.: On developing fourth-order optimal families of methods for multiple roots and their dynamics. Appl. Math. Comput. 265(15), 520–532 (2015)MathSciNetGoogle Scholar
  6. 6.
    Behl, R., Cordero, A., Motsa, S.S., Torregrosa, J.R., Kanwar, V.: An optimal fourth-order family of methods for multiple roots and its dynamics. Numer. Algor. 71(4), 775–796 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chicharro, F.I., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameters planes of iterative families and methods. Sci World J 2013(Article ID 780153), 11 (2013)Google Scholar
  8. 8.
    Li, S., Liao, X., Cheng, L.: A new fourth-order iterative method for finding multiple roots of nonlinear equations. Appl. Math. Comput. 215, 1288–1292 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Li, S., Cheng, L., Neta, B.: Some fourth-order nonlinear solvers with closed formulae for multiple roots. Comput. Math. Appl. 59, 126–135 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Neta, B., Chun, C., Scott, M.: On the development of iterative methods for multiple roots. Appl. Math. Comput. 224, 358–361 (2013)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Sharma, J.R., Sharma, R.: Modified Jarratt method for computing multiple roots. Appl. Math. Comput. 217, 878–881 (2010)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Zhou, X., Chen, X., Song, Y.: Constructing higher-order methods for obtaining the multiple roots of nonlinear equations. Comput. Appl. Math. 235, 4199–4206 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Zhou, X., Chen, X., Song, Y.: Families of third and fourth order methods for multiple roots of nonlinear equations. Appl. Math. Comput. 219, 6030–6038 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Neta, B.: Extension of Murakami’s high-order non-linear solver to multiple roots. Int. J. Comput. Math. 87(5), 1023–1031 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    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
  16. 16.
    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
  17. 17.
    Artidiello, S., Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Two weighted eight-order classes of iterative root-finding methods. Int. J. Comput. Math. 92(9), 1790–1805 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Shacham, M.: Numerical solution of constrained nonlinear algebraic equations. Int. J. Numer. Method Eng. 23, 1455–1481 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190(1), 686–698 (2007)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Computer SciencesUniversity of KwaZulu-NatalScottsvilleSouth Africa
  2. 2.Instituto Universitario de Matemática MultidisciplinarUniversitat Politècnica de ValènciaValènciaSpain
  3. 3.Mathematics DepartmentUniversity of SwazilandKwaluseniSwaziland

Personalised recommendations