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On developing fourth-order optimal families of methods for multiple roots and their dynamics

Author

Listed:
  • Behl, Ramandeep
  • Cordero, Alicia
  • Motsa, S.S.
  • Torregrosa, Juan R.

Abstract

There are few optimal fourth-order methods for solving nonlinear equations when the multiplicity m of the required root is known in advance. Therefore, the first focus of this paper is on developing new fourth-order optimal families of iterative methods by a simple and elegant way. Computational and theoretical properties are fully studied along with a main theorem describing the convergence analysis. Another main focus of this paper is the dynamical analysis of the rational map associated with our proposed class for multiple roots; as far as we know, there are no deep study of this kind on iterative methods for multiple roots. Further, using Mathematica with its high precision compatibility, a variety of concrete numerical experiments and relevant results are extensively treated to confirm the theoretical development.

Suggested Citation

  • Behl, Ramandeep & Cordero, Alicia & Motsa, S.S. & Torregrosa, Juan R., 2015. "On developing fourth-order optimal families of methods for multiple roots and their dynamics," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 520-532.
    • Handle: RePEc:eee:apmaco:v:265:y:2015:i:c:p:520-532
    DOI: 10.1016/j.amc.2015.05.004
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    References listed on IDEAS

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    1. Neta, Beny & Chun, Changbum, 2014. "Basins of attraction for several optimal fourth order methods for multiple roots," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 103(C), pages 39-59.
    2. 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.
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    Cited by:

    1. Geum, Young Hee & Kim, Young Ik & Magreñán, Á. Alberto, 2016. "A biparametric extension of King’s fourth-order methods and their dynamics," Applied Mathematics and Computation, Elsevier, vol. 282(C), pages 254-275.
    2. Ramandeep Behl & Munish Kansal & Mehdi Salimi, 2020. "Modified King’s Family for Multiple Zeros of Scalar Nonlinear Functions," Mathematics, MDPI, vol. 8(5), pages 1-17, May.
    3. Fiza Zafar & Alicia Cordero & Juan R. Torregrosa, 2018. "An Efficient Family of Optimal Eighth-Order Multiple Root Finders," Mathematics, MDPI, vol. 6(12), pages 1-16, December.
    4. Samundra Regmi & Ioannis K. Argyros & Santhosh George, 2024. "Convergence of High-Order Derivative-Free Algorithms for the Iterative Solution of Systems of Not Necessarily Differentiable Equations," Mathematics, MDPI, vol. 12(5), pages 1-13, February.
    5. Lee, Min-Young & Ik Kim, Young & Alberto Magreñán, Á., 2017. "On the dynamics of a triparametric family of optimal fourth-order multiple-zero finders with a weight function of the principal mth root of a function-to function ratio," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 564-590.
    6. Min-Young Lee & Young Ik Kim, 2020. "Bifurcations along the Boundary Curves of Red Fixed Components in the Parameter Space for Uniparametric, Jarratt-Type Simple-Root Finders," Mathematics, MDPI, vol. 8(1), pages 1-13, January.
    7. Abhimanyu Kumar & Dharmendra K. Gupta & Eulalia Martínez & Sukhjit Singh, 2018. "Convergence of a Two-Step Iterative Method for Nondifferentiable Operators in Banach Spaces," Complexity, Hindawi, vol. 2018, pages 1-11, May.
    8. Ramandeep Behl & Eulalia Martínez & Fabricio Cevallos & Diego Alarcón, 2019. "A Higher Order Chebyshev-Halley-Type Family of Iterative Methods for Multiple Roots," Mathematics, MDPI, vol. 7(4), pages 1-12, April.
    9. Ramandeep Behl & Sonia Bhalla & Ángel Alberto Magreñán & Alejandro Moysi, 2021. "An Optimal Derivative Free Family of Chebyshev–Halley’s Method for Multiple Zeros," Mathematics, MDPI, vol. 9(5), pages 1-19, March.
    10. Ramandeep Behl & Sonia Bhalla & Eulalia Martínez & Majed Aali Alsulami, 2021. "Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions," Mathematics, MDPI, vol. 9(11), pages 1-14, May.
    11. Young Hee Geum & Young Ik Kim, 2020. "Computational Bifurcations Occurring on Red Fixed Components in the λ -Parameter Plane for a Family of Optimal Fourth-Order Multiple-Root Finders under the Möbius Conjugacy Map," Mathematics, MDPI, vol. 8(5), pages 1-17, May.
    12. Young Hee Geum & Young Ik Kim, 2019. "On Locating and Counting Satellite Components Born along the Stability Circle in the Parameter Space for a Family of Jarratt-Like Iterative Methods," Mathematics, MDPI, vol. 7(9), pages 1-16, September.
    13. Francisco I. Chicharro & Rafael A. Contreras & Neus Garrido, 2020. "A Family of Multiple-Root Finding Iterative Methods Based on Weight Functions," Mathematics, MDPI, vol. 8(12), pages 1-17, December.

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