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Influence of the Center Condition on the Two-Step Secant Method

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  • Abhimanyu Kumar
  • D. K. Gupta
  • Shwetabh Srivastava

Abstract

The aim of this paper is to present a new improved semilocal and local convergence analysis for two-step secant method to approximate a locally unique solution of a nonlinear equation in Banach spaces. This study is important because starting points play an important role in the convergence of an iterative method. We have used a combination of Lipschitz and center-Lipschitz conditions on the Fréchet derivative instead of only Lipschitz condition. A comparison is established on different types of center conditions and the influence of our approach is shown through the numerical examples. In comparison to some earlier study, it gives an improved domain of convergence along with the precise error bounds. Finally, some numerical examples including nonlinear elliptic differential equations and integral equations validate the efficacy of our approach.

Suggested Citation

  • Abhimanyu Kumar & D. K. Gupta & Shwetabh Srivastava, 2017. "Influence of the Center Condition on the Two-Step Secant Method," International Journal of Analysis, Hindawi, vol. 2017, pages 1-9, September.
    • Handle: RePEc:hin:ijanal:7364236
    DOI: 10.1155/2017/7364236
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    References listed on IDEAS

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    1. Magreñán, Á. Alberto & Argyros, Ioannis K., 2016. "New improved convergence analysis for the secant method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 119(C), pages 161-170.
    2. Argyros, Ioannis K. & Cordero, Alicia & Magreñán, Alberto & Torregrosa, Juan R., 2015. "On the convergence of a damped Newton-like method with modified right hand side vector," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 927-936.
    3. Argyros, Ioannis K. & Magreñán, Á. Alberto, 2015. "Expanding the applicability of the Secant method under weaker conditions," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 1000-1012.
    4. Magreñán, Á. Alberto & Argyros, Ioannis K., 2015. "New semilocal and local convergence analysis for the Secant method," Applied Mathematics and Computation, Elsevier, vol. 262(C), pages 298-307.
    5. Argyros, Ioannis K. & Cordero, Alicia & Alberto Magreñán, Á. & Torregrosa, J.R., 2015. "On the convergence of a Damped Secant method with modified right-hand side vector," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 315-323.
    6. Singh, Sukhjit & Gupta, Dharmendra Kumar & Martínez, E. & Hueso, José L., 2016. "Semilocal and local convergence of a fifth order iteration with Fréchet derivative satisfying Hölder condition," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 266-277.
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    Cited by:

    1. Ioannis K. Argyros & Neha Gupta & J. P. Jaiswal, 2019. "Extending the Applicability of a Two-Step Chord-Type Method for Non-Differentiable Operators," Mathematics, MDPI, vol. 7(9), pages 1-8, September.

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