A general class of four parametric with- and without memory iterative methods for nonlinear equations

  • Fiza ZafarEmail author
  • Alicia Cordero
  • Juan R. Torregrosa
  • Aneeqa Rafi
Original Paper


In this paper, we have constructed a derivative-free weighted eighth-order iterative class of methods with and without-memory for solving nonlinear equations. These methods are optimal as they satisfy Kung–Traub’s conjecture. We have used four accelerating parameters, univariate and multivariate weight functions at the second and third step of the method respectively. This family of schemes is converted into with-memory one by approximating the parameters using Newton’s interpolating polynomials of appropriate degree to increase the order of convergence to 15.51560 and the efficiency index is nearly two. Numerical and dynamical comparison of our methods is done with some recent methods of the same order applying them on some applied problems from chemical engineering, such as fractional conversion in a chemical reactor. The stability of the proposed schemes and their comparison with existing ones is made by using dynamical planes of the different methods, showing the wideness of the sets of converging initial estimations for all the test functions. The proposed schemes show to have good stability properties, as in their eighth-order version as well as in the case of methods with memory.


Nonlinear equations Iterative method with and without memory Basin of attraction Order of convergence 

Mathematics Subject Classification




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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Centre for Advanced Studies in Pure and Applied MathematicsBahauddin Zakariya UniversityMultanPakistan
  2. 2.Institute for Multidisciplinary MathematicsUniversitat Politecnica de ValenciaValenciaSpain

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