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A New Nonlinear Ninth-Order Root-Finding Method with Error Analysis and Basins of Attraction

Author

Listed:
  • Sania Qureshi

    (Department of Mathematics, Near East University TRNC, Mersin 99138, Turkey
    Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro 76062, Pakistan
    These authors contributed equally to this work.)

  • Higinio Ramos

    (Scientific Computing Group, Universidad de Salamanca, 37008 Salamanca, Spain
    Escuela Politécnica Superior de Zamora, Avda. Requejo 33, 49029 Zamora, Spain
    These authors contributed equally to this work.)

  • Abdul Karim Soomro

    (Institute of Mathematics and Computer Sciences, University of Sindh, Jamshoro 76062, Pakistan
    These authors contributed equally to this work.)

Abstract

Nonlinear phenomena occur in various fields of science, business, and engineering. Research in the area of computational science is constantly growing, with the development of new numerical schemes or with the modification of existing ones. However, such numerical schemes, objectively need to be computationally inexpensive with a higher order of convergence. Taking into account these demanding features, this article attempted to develop a new three-step numerical scheme to solve nonlinear scalar and vector equations. The scheme was shown to have ninth order convergence and requires six function evaluations per iteration. The efficiency index is approximately 1.4422, which is higher than the Newton’s scheme and several other known optimal schemes. Its dependence on the initial estimates was studied by using real multidimensional dynamical schemes, showing its stable behavior when tested upon some nonlinear models. Based on absolute errors, the number of iterations, the number of function evaluations, preassigned tolerance, convergence speed, and CPU time (sec), comparisons with well-known optimal schemes available in the literature showed a better performance of the proposed scheme. Practical models under consideration include open-channel flow in civil engineering, Planck’s radiation law in physics, the van der Waals equation in chemistry, and the steady-state of the Lorenz system in meteorology.

Suggested Citation

  • Sania Qureshi & Higinio Ramos & Abdul Karim Soomro, 2021. "A New Nonlinear Ninth-Order Root-Finding Method with Error Analysis and Basins of Attraction," Mathematics, MDPI, vol. 9(16), pages 1-19, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:16:p:1996-:d:618621
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    References listed on IDEAS

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    1. Juan R. Torregrosa & Ioannis K. Argyros & Changbum Chun & Alicia Cordero & Fazlollah Soleymani, 2014. "Iterative Methods for Nonlinear Equations or Systems and Their Applications 2014," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-2, November.
    2. Huijuan Chen & Xintao Zheng, 2020. "Improved Newton Iterative Algorithm for Fractal Art Graphic Design," Complexity, Hindawi, vol. 2020, pages 1-11, November.
    3. Abro, Hameer Akhtar & Shaikh, Muhammad Mujtaba, 2019. "A new time-efficient and convergent nonlinear solver," Applied Mathematics and Computation, Elsevier, vol. 355(C), pages 516-536.
    4. Amir Naseem & M. A. Rehman & Thabet Abdeljawad & Kaleem R. Kazmi, 2020. "Higher-Order Root-Finding Algorithms and Their Basins of Attraction," Journal of Mathematics, Hindawi, vol. 2020, pages 1-11, November.
    5. Muhammad Aslam Noor & Khalida Inayat Noor & Eisa Al-Said & Muhammad Waseem, 2010. "Some New Iterative Methods for Nonlinear Equations," Mathematical Problems in Engineering, Hindawi, vol. 2010, pages 1-12, January.
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