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An Efficient Family of Optimal Eighth-Order Multiple Root Finders

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  • Fiza Zafar

    (Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan
    Instituto de Matemáticas Multidisciplinar, Universitat Politènica de València, 46022 València, Spain)

  • Alicia Cordero

    (Instituto de Matemáticas Multidisciplinar, Universitat Politènica de València, 46022 València, Spain)

  • Juan R. Torregrosa

    (Instituto de Matemáticas Multidisciplinar, Universitat Politènica de València, 46022 València, Spain)

Abstract

Finding a repeated zero for a nonlinear equation f ( x ) = 0 , f : I ⊆ R → R has always been of much interest and attention due to its wide applications in many fields of science and engineering. Modified Newton’s method is usually applied to solve this kind of problems. Keeping in view that very few optimal higher-order convergent methods exist for multiple roots, we present a new family of optimal eighth-order convergent iterative methods for multiple roots with known multiplicity involving a multivariate weight function. The numerical performance of the proposed methods is analyzed extensively along with the basins of attractions. Real life models from life science, engineering, and physics are considered for the sake of comparison. The numerical experiments and dynamical analysis show that our proposed methods are efficient for determining multiple roots of nonlinear equations.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:12:p:310-:d:188848
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    References listed on IDEAS

    as
    1. 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.
    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. 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.
    4. 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.
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