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On a Bi-Parametric Family of Fourth Order Composite Newton–Jarratt Methods for Nonlinear Systems

Author

Listed:
  • Janak Raj Sharma

    (Department of Mathematics, Sant Longowal Institute of Engineering and Technology Longowal, Sangrur 148106, India)

  • Deepak Kumar

    (Department of Mathematics, Sant Longowal Institute of Engineering and Technology Longowal, Sangrur 148106, India)

  • Ioannis K. Argyros

    (Department of Mathematics Sciences, Cameron University, Lawton, OK 73505, USA)

  • Ángel Alberto Magreñán

    (Departamento de Matemáticas y Computación, Universidad de La Rioja, 26004 Logroño, La Rioja, Spain)

Abstract

We present a new two-parameter family of fourth-order iterative methods for solving systems of nonlinear equations. The scheme is composed of two Newton–Jarratt steps and requires the evaluation of one function and two first derivatives in each iteration. Convergence including the order of convergence, the radius of convergence, and error bounds is presented. Theoretical results are verified through numerical experimentation. Stability of the proposed class is analyzed and presented by means of using new dynamics tool, namely, the convergence plane. Performance is exhibited by implementing the methods on nonlinear systems of equations, including those resulting from the discretization of the boundary value problem. In addition, numerical comparisons are made with the existing techniques of the same order. Results show the better performance of the proposed techniques than the existing ones.

Suggested Citation

  • Janak Raj Sharma & Deepak Kumar & Ioannis K. Argyros & Ángel Alberto Magreñán, 2019. "On a Bi-Parametric Family of Fourth Order Composite Newton–Jarratt Methods for Nonlinear Systems," Mathematics, MDPI, vol. 7(6), pages 1-27, May.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:6:p:492-:d:235453
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    References listed on IDEAS

    as
    1. Artidiello, S. & Cordero, Alicia & Torregrosa, Juan R. & Vassileva, M.P., 2017. "Design and multidimensional extension of iterative methods for solving nonlinear problems," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 194-203.
    2. Abbasbandy, Saeid & Bakhtiari, Parisa & Cordero, Alicia & Torregrosa, Juan R. & Lotfi, Taher, 2016. "New efficient methods for solving nonlinear systems of equations with arbitrary even order," Applied Mathematics and Computation, Elsevier, vol. 287, pages 94-103.
    3. Argyros, Ioannis K. & Magreñán, Á. Alberto, 2015. "On the convergence of an optimal fourth-order family of methods and its dynamics," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 336-346.
    4. Sharma, Janak Raj & Sharma, Rajni & Kalra, Nitin, 2015. "A novel family of composite Newton–Traub methods for solving systems of nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 520-535.
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