IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i1p51-d304212.html
   My bibliography  Save this article

Bifurcations along the Boundary Curves of Red Fixed Components in the Parameter Space for Uniparametric, Jarratt-Type Simple-Root Finders

Author

Listed:
  • Min-Young Lee

    (Department of Applied Mathematics, Dankook University, Cheonan 330-714, Korea)

  • Young Ik Kim

    (Department of Applied Mathematics, Dankook University, Cheonan 330-714, Korea)

Abstract

Bifurcations have been studied with an extensive analysis of boundary curves of red, fixed components in the parametric space for a uniparametric family of simple-root finders under the Möbius conjugacy map applied to a quadratic polynomial. An elementary approach from the perspective of a plane curve theory properly describes the geometric figures resembling a circle or cardioid to characterize the underlying boundary curves that are parametrically expressed. Moreover, exact bifurcation points for satellite components on the boundaries have been found, according to the fact that the tangent line at a bifurcation point simultaneously touches the red fixed component and the satellite component. Computational experiments implemented with examples well reflect the significance of the theoretical backgrounds pursued in this paper.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:1:p:51-:d:304212
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/1/51/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/8/1/51/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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. 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.
    3. Young Ik Kim & Young Hee Geum, 2013. "A Two-Parameter Family of Fourth-Order Iterative Methods with Optimal Convergence for Multiple Zeros," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-7, February.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. 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.
    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. 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.
    6. 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.
    7. Min-Young Lee & Young Ik Kim & Beny Neta, 2019. "A Generic Family of Optimal Sixteenth-Order Multiple-Root Finders and Their Dynamics Underlying Purely Imaginary Extraneous Fixed Points," Mathematics, MDPI, vol. 7(6), pages 1-26, June.
    8. Young Hee Geum & Young Ik Kim & Beny Neta, 2018. "Developing an Optimal Class of Generic Sixteenth-Order Simple-Root Finders and Investigating Their Dynamics," Mathematics, MDPI, vol. 7(1), pages 1-32, December.
    9. 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.
    10. 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.
    11. Young-Hee Geum & Young-Ik Kim, 2021. "Computational Geometry of Period-3 Hyperbolic Components in the Mandelbrot Set," Mathematics, MDPI, vol. 9(19), pages 1-15, October.
    12. 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.
    13. Cordero, Alicia & Soleymani, Fazlollah & Torregrosa, Juan R. & Haghani, F. Khaksar, 2017. "A family of Kurchatov-type methods and its stability," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 264-279.
    14. 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.
    15. 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.
    16. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:8:y:2020:i:1:p:51-:d:304212. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.