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One Parameter Optimal Derivative-Free Family to Find the Multiple Roots of Algebraic Nonlinear Equations

Author

Listed:
  • Munish Kansal

    (School of Mathematics, Thapar Institute of Engineering and Technology, Punjab 147004, India)

  • Ali Saleh Alshomrani

    (Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia)

  • Sonia Bhalla

    (Department of Mathematics, Chandigarh University, Gharuan, Mohali 140413, India)

  • Ramandeep Behl

    (Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia)

  • Mehdi Salimi

    (Department of Mathematics & Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada
    Center for Dynamics, Faculty of Mathematics, Technische Universität Dresden, 01062 Dresden, Germany)

Abstract

In this study, we construct the one parameter optimal derivative-free iterative family to find the multiple roots of an algebraic nonlinear function. Many researchers developed the higher order iterative techniques by the use of the new function evaluation or the first-order or second-order derivative of functions to evaluate the multiple roots of a nonlinear equation. However, the evaluation of the derivative at each iteration is a cumbersome task. With this motivation, we design the second-order family without the utilization of the derivative of a function and without the evaluation of the new function. The proposed family is optimal as it satisfies the convergence order of Kung and Traub’s conjecture. Here, we use one parameter a for the construction of the scheme, and for a = 1 , the modified Traub method is its a special case. The order of convergence is analyzed by Taylor’s series expansion. Further, the efficiency of the suggested family is explored with some numerical tests. The obtained results are found to be more efficient than earlier schemes. Moreover, the basin of attraction of the proposed and earlier schemes is also analyzed.

Suggested Citation

  • Munish Kansal & Ali Saleh Alshomrani & Sonia Bhalla & Ramandeep Behl & Mehdi Salimi, 2020. "One Parameter Optimal Derivative-Free Family to Find the Multiple Roots of Algebraic Nonlinear Equations," Mathematics, MDPI, vol. 8(12), pages 1-15, December.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:12:p:2223-:d:461836
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    References listed on IDEAS

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    1. Geum, Young Hee & Kim, Young Ik & Neta, Beny, 2015. "A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 387-400.
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    Cited by:

    1. Himani Arora & Alicia Cordero & Juan R. Torregrosa & Ramandeep Behl & Sattam Alharbi, 2022. "Derivative-Free Iterative Schemes for Multiple Roots of Nonlinear Functions," Mathematics, MDPI, vol. 10(9), pages 1-13, May.

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