IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v265y2015icp520-532.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300315006025
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2015.05.004?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    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. Petković, I. & Herceg, Ð., 2017. "Symbolic computation and computer graphics as tools for developing and studying new root-finding methods," Applied Mathematics and Computation, Elsevier, vol. 295(C), pages 95-113.
    2. Geum, Young Hee & Kim, Young Ik & Neta, Beny, 2016. "A sixth-order family of three-point modified Newton-like multiple-root finders and the dynamics behind their extraneous fixed points," Applied Mathematics and Computation, Elsevier, vol. 283(C), pages 120-140.
    3. 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.
    4. Chun, Changbum & Neta, Beny, 2015. "Basins of attraction for several third order methods to find multiple roots of nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 129-137.
    5. 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.
    6. Geum, Young Hee & Kim, Young Ik & Neta, Beny, 2015. "A class of two-point sixth-order multiple-zero finders of modified double-Newton type and their dynamics," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 387-400.
    7. Chun, Changbum & Neta, Beny, 2016. "Comparison of several families of optimal eighth order methods," Applied Mathematics and Computation, Elsevier, vol. 274(C), pages 762-773.
    8. Chun, Changbum & Neta, Beny, 2015. "Comparing the basins of attraction for Kanwar–Bhatia–Kansal family to the best fourth order method," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 277-292.
    9. Argyros, Ioannis K. & Kansal, Munish & Kanwar, Vinay & Bajaj, Sugandha, 2017. "Higher-order derivative-free families of Chebyshev–Halley type methods with or without memory for solving nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 224-245.
    10. Chun, Changbum & Neta, Beny, 2015. "Basins of attraction for Zhou–Chen–Song fourth order family of methods for multiple roots," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 109(C), pages 74-91.
    11. Ioannis K. Argyros & Ángel Alberto Magreñán & Lara Orcos & Íñigo Sarría, 2019. "Unified Local Convergence for Newton’s Method and Uniqueness of the Solution of Equations under Generalized Conditions in a Banach Space," Mathematics, MDPI, vol. 7(5), pages 1-13, May.
    12. Chun, Changbum & Neta, Beny, 2016. "An analysis of a Khattri’s 4th order family of methods," Applied Mathematics and Computation, Elsevier, vol. 279(C), pages 198-207.
    13. Ramandeep Behl & Ioannis K. Argyros, 2020. "Local Convergence for Multi-Step High Order Solvers under Weak Conditions," Mathematics, MDPI, vol. 8(2), pages 1-14, February.
    14. Behl, Ramandeep & Cordero, Alicia & Motsa, Sandile S. & Torregrosa, Juan R., 2015. "Construction of fourth-order optimal families of iterative methods and their dynamics," Applied Mathematics and Computation, Elsevier, vol. 271(C), pages 89-101.
    15. 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.
    16. García Calcines, José M. & Gutiérrez, José M. & Hernández Paricio, Luis J. & Rivas Rodríguez, M. Teresa, 2015. "Graphical representations for the homogeneous bivariate Newton’s method," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 988-1006.
    17. Sharifi, Somayeh & Salimi, Mehdi & Siegmund, Stefan & Lotfi, Taher, 2016. "A new class of optimal four-point methods with convergence order 16 for solving nonlinear equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 119(C), pages 69-90.
    18. Geum, Young Hee & Kim, Young Ik & Neta, Beny, 2017. "A family of optimal quartic-order multiple-zero finders with a weight function of the principal kth root of a derivative-to-derivative ratio and their basins of attraction," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 136(C), pages 1-21.
    19. 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.
    20. Geum, Young Hee & Kim, Young Ik & Neta, Beny, 2015. "On developing a higher-order family of double-Newton methods with a bivariate weighting function," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 277-290.

    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:eee:apmaco:v:265:y:2015:i:c:p:520-532. 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: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    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.