SeMA Journal

pp 1–21 | Cite as

Optimal fourth order methods with its multi-step version for nonlinear equation and their Basins of attraction

  • Parimala Sivakumar
  • Kalyanasundaram Madhu
  • Jayakumar JayaramanEmail author


Construction of iterative processes without memory, which is optimal according to the Kung–Traub hypothesis is considered in this paper. A new class of method by improving Ezquerro et al. method having three function evaluations per iteration, which reaches the optimal order four is discussed. The efficiency index of this method is found to be 1.587 which is better than the efficiency index of Newton’s method (1.414) and equals the efficiency of Jarratt and Chun’s methods. This procedure also provides an n-step family of methods using n + 1 function evaluations per iteration cycle to possess the order (2r + 4). The theoretical proof of the main contributions are given and numerical examples are included to confirm the convergence order of the presented methods. The extraneous fixed points of the presented fourth order method and other existing fourth order methods are discussed. Basins of attraction study is also done for the presented methods and few other methods. We apply the new methods to find the optimal launch angle in a projectile motion problem as an application.


Non-linear equation Multi-point iterations Optimal order Higher order method Kung–Traub’s conjecture Extraneous fixed points 

Mathematics Subject Classification

65H05 65D05 41A25 



The authors would like to thank the editor and referees for their valuable comments and suggestions.


  1. 1.
    Amat, S., Busquier, S., Plaza, S.: Dynamics of a family of third-order iterative methods that do not require using second derivatives. Appl. Math. Comp. 154, 735–746 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Amat, S., Busquier, S., Plaza, S.: Review of some iterative root-finding methods from a dynamical point of view. Sci. Ser. A Math. Sci. 10, 3–35 (2004)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Amat, S., Busquier, S., Plaza, S.: Dynamics of the King and Jarratt iterations. Aequ. Math. 69, 212–223 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Babajee, D.K.R., Madhu, K.: Comparing two techniques for developing higher order two-point iterative methods for solving quadratic equations. SeMA J. 1, 1 (2019). MathSciNetGoogle Scholar
  5. 5.
    Beardon, A.F.: Iteration of rational functions. Springer, New York (1991)CrossRefzbMATHGoogle Scholar
  6. 6.
    Chapra, S.C., Canale, R.P.: Numerical methods for engineers. McGraw-Hill, New York (1988)Google Scholar
  7. 7.
    Chicharro, F., Cordero, A., Gutierrez, J.M., Torregrosa, J.R.: Complex dynamics of derivative-free methods for nonlinear equations. Appl. Math. Comput. 219(12), 7023–7035 (2013)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chun, C.: A family of composite fourth-order iterative methods for solving nonlinear equations. Appl. Math. Comput. 187, 951–956 (2007)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chun, C., Lee, M.Y., Neta, B., Dzunic, J.: On optimal fourth-order iterative methods free from second derivative and their dynamics. Appl. Math. Comput. 218, 6427–6438 (2012)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Cordero, A., Hueso, J.L., Martinez, E., Torregrosa, J.R.: New modifcations of Potra–Ptak’s method with optimal fourth and eighth orders of convergence. J. Comput. Appl. Math. 234, 2969–2976 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cordero, A., Torregrosa, J.R., Vindel, P.: Dynamics of a family of Chebyshev–Halley type methods. Appl. Math. Comput. 219(16), 8568–8583 (2013)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Curry, J.H., Garnett, L., Sullivan, D.: On the iteration of a rational function: computer experiments with Newton’s method. Commun. Math. Phys. 91, 267–277 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ezquerro, J.A., Hernandez, M.A., Salanova, M.A.: A new family of mulitipoint methods of second order. Numer. Funct. Anal. Opt. 19(5–6), 499–512 (1998)CrossRefzbMATHGoogle Scholar
  14. 14.
    Halley, E.: A new, exact and easy method for finding the roots of equations generally and that without any previous reduction. Phil. Trans. Roy. Soc. London. 18, 136–148 (1694)Google Scholar
  15. 15.
    Jarratt, P.: Some fourth order multipoint iterative methods for solving equations. Math. Comp. 20, 434–437 (1966)CrossRefzbMATHGoogle Scholar
  16. 16.
    Jarratt, P.: Some efficient fourth order multipoint methods for solving equations. BIT 9, 119–124 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kantrowitz, R., Neumann, M.M.: Some real analysis behind optimization of projectile motion. Mediterr. J. Math. 11(4), 1081–1097 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    King, R.F.: A family of fourth-order methods for solving nonlinear equations. SIAM J. Numer. Anal. 10, 876–879 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kneisl, K.: Julia sets for the super-Newton’s method, Cauchy’s method and Halley’s method. Chaos 11(2), 359–370 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kou, J., Li, Y., Wang, X.: A composite fourth-order iterative method for solving non-linear equations. Appl. Math. Comp. 184, 471–475 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach. 21(4), 643–651 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Madhu, K., Jayaraman, J.: Higher order methods for nonlinear equations and their basins of attraction. Mathematics 4, 22 (2016)CrossRefzbMATHGoogle Scholar
  23. 23.
    Neta, B.: Numerical methods for the solution of equations. NetA-Sof, Monterey (1983)zbMATHGoogle Scholar
  24. 24.
    Ostrowski, A.M.: Solutions of equations and system of equations. Academic Press, New York (1960)Google Scholar
  25. 25.
    Petkovic, M.S., Neta, B., Petkovic, L.D., Dzunic, J.: Multipoint methods for solving nonlinear equations. Academic Press (an imprint of Elsevier), Waltham, MA (2013)Google Scholar
  26. 26.
    Scott, M., Neta, B., Chun, C.: Basin attractors for various methods. Appl. Math. Comput. 218, 2584–2599 (2011)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Sharifi, M., Babajee, D., Soleymani, F.: Finding the solution of nonlinear equations by a class of optimal methods. Comput. Math. Appl. 63, 764–774 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Soleymani, F., Babajee, D., Sharifi, M.: Modified Jarratt method without memory with twelfth-order convergence. Ann. Univ. Craiova Math. Comput. Sci. Ser. 39, 21–34 (2012)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Traub, J.F.: Iterative methods for the solution of equations. Chelsea Publishing Company, New York (1977)Google Scholar
  30. 30.
    Vrscay, E.R.: Julia sets and Mandelbrot-like sets associated with higher order Schroder rational iteration functions: a computer assisted study. Math Comput 46, 151–169 (1986)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Vrscay, E.R., Gilbert, W.J.: Extraneous fxed points, basin boundaries and chaotic dynamics for schroder and konig rational iteration functions. Numer. Math. 52, 1–16 (1987)CrossRefzbMATHGoogle Scholar

Copyright information

© Sociedad Española de Matemática Aplicada 2019

Authors and Affiliations

  1. 1.Department of MathematicsPondicherry Engineering CollegePondicherryIndia
  2. 2.Department of MathematicsSaveetha Engineering CollegeChennaiIndia

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