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Using Matrix Eigenvalues to Construct an Iterative Method with the Highest Possible Efficiency Index Two

Author

Listed:
  • Malik Zaka Ullah

    (Mathematical Modeling and Applied Computation (MMAC) Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia)

  • Vali Torkashvand

    (Member of Young Researchers and Elite Club, Shahr-e-Qods Branch, Islamic Azad University, Tehran 37515-374, Iran)

  • Stanford Shateyi

    (Department of Mathematics and Applied Mathematics, School of Mathematical and Natural Sciences, University of Venda, P. Bag X5050, Thohoyandou 0950, South Africa)

  • Mir Asma

    (Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia)

Abstract

In this paper, we first derive a family of iterative schemes with fourth order. A weight function is used to maintain its optimality. Then, we transform it into methods with several self-accelerating parameters to reach the highest possible convergence rate 8. For this aim, we employ the property of the eigenvalues of the matrices and the technique with memory. Solving several nonlinear test equations shows that the proposed variants have a computational efficiency index of two (maximum amount possible) in practice.

Suggested Citation

  • Malik Zaka Ullah & Vali Torkashvand & Stanford Shateyi & Mir Asma, 2022. "Using Matrix Eigenvalues to Construct an Iterative Method with the Highest Possible Efficiency Index Two," Mathematics, MDPI, vol. 10(9), pages 1-15, April.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:9:p:1370-:d:797451
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    References listed on IDEAS

    as
    1. Soleymani, F. & Lotfi, T. & Tavakoli, E. & Khaksar Haghani, F., 2015. "Several iterative methods with memory using self-accelerators," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 452-458.
    2. Soheili, Ali R. & Amini, Mohammad & Soleymani, Fazlollah, 2019. "A family of Chaplygin-type solvers for Itô stochastic differential equations," Applied Mathematics and Computation, Elsevier, vol. 340(C), pages 296-304.
    3. Xiaofeng Wang & Mingming Zhu, 2020. "Two Iterative Methods with Memory Constructed by the Method of Inverse Interpolation and Their Dynamics," Mathematics, MDPI, vol. 8(7), pages 1-12, July.
    Full references (including those not matched with items on IDEAS)

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