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Symbolic computation and computer graphics as tools for developing and studying new root-finding methods

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  • Petković, I.
  • Herceg, Ð.

Abstract

Many very difficult problems in applied mathematics and other scientific disciplines cannot be solved without powerful computational systems, such as symbolic computation and computer graphics. In this paper we construct two new families of the fourth order iterative methods for finding a multiple real or complex zero of a given function. For developing these methods, a recurrent formula for generating iterative methods of higher order for solving nonlinear equations is applied and implemented by symbolic computation through several programs in computer algebra system Mathematica. Symbolic computation was the only tool for solving the considered complex problem since it provides handling and manipulating complex mathematical expressions and other mathematical objects. The properties of the proposed rapidly convergent methods are illustrated by several numerical examples. To examine the convergence behavior of the presented methods, we also give the dynamic study of these methods using basins of attraction. Such a methodology, besides a visualization of iterative processes, deliveries very important features on iterations including running CPU time and average number of iterations, as a function of starting points. The program for plotting basins of attraction in Mathematica is included.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:295:y:2017:i:c:p:95-113
    DOI: 10.1016/j.amc.2016.09.025
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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. 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.
    3. 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.
    4. 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.
    5. 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.
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    Cited by:

    1. Proinov, Petko D. & Ivanov, Stoil I. & Petković, Miodrag S., 2019. "On the convergence of Gander’s type family of iterative methods for simultaneous approximation of polynomial zeros," Applied Mathematics and Computation, Elsevier, vol. 349(C), pages 168-183.

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