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A class of optimal eighth-order derivative-free methods for solving the Danchick–Gauss problem

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  • Andreu, Carlos
  • Cambil, Noelia
  • Cordero, Alicia
  • Torregrosa, Juan R.

Abstract

A derivative-free optimal eighth-order family of iterative methods for solving nonlinear equations is constructed using weight functions approach with divided first order differences. Its performance, along with several other derivative-free methods, is studied on the specific problem of Danchick’s reformulation of Gauss’ method of preliminary orbit determination. Numerical experiments show that such derivative-free, high-order methods offer significant advantages over both, the classical and Danchick’s Newton approach.

Suggested Citation

  • Andreu, Carlos & Cambil, Noelia & Cordero, Alicia & Torregrosa, Juan R., 2014. "A class of optimal eighth-order derivative-free methods for solving the Danchick–Gauss problem," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 237-246.
    • Handle: RePEc:eee:apmaco:v:232:y:2014:i:c:p:237-246
    DOI: 10.1016/j.amc.2014.01.056
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    References listed on IDEAS

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    1. F. Soleymani & S. Shateyi, 2012. "Two Optimal Eighth-Order Derivative-Free Classes of Iterative Methods," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-14, November.
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    Cited by:

    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. 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.
    3. 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.
    4. 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.
    5. 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.
    6. Danchick, Roy, 2015. "Simplified existence and uniqueness conditions for the zeros and the concavity of the F and G functions of improved Gauss orbit determination from two position vectors," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 279-287.

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