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Third‐Order Newton‐Type Methods Combined with Vector Extrapolation for Solving Nonlinear Systems

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  • Wen Zhou
  • Jisheng Kou

Abstract

We present a third‐order method for solving the systems of nonlinear equations. This method is a Newton‐type scheme with the vector extrapolation. We establish the local and semilocal convergence of this method. Numerical results show that the composite method is more robust and efficient than a number of Newton‐type methods with the other vector extrapolations.

Suggested Citation

  • Wen Zhou & Jisheng Kou, 2014. "Third‐Order Newton‐Type Methods Combined with Vector Extrapolation for Solving Nonlinear Systems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:601745
    DOI: 10.1155/2014/601745
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    References listed on IDEAS

    as
    1. Jinsu He & Jinhua Wang & Jen-Chih Yao, 2013. "Local Convergence of Newton’s Method on Lie Groups and Uniqueness Balls," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    2. Jinsu He & Jinhua Wang & Jen-Chih Yao, 2013. "Local Convergence of Newton’s Method on Lie Groups and Uniqueness Balls," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-9, November.
    3. Jisheng Kou & Shuyu Sun & Bo Yu, 2011. "Multiscale Time‐Splitting Strategy for Multiscale Multiphysics Processes of Two‐Phase Flow in Fractured Media," Journal of Applied Mathematics, John Wiley & Sons, vol. 2011(1).
    4. S. Amat & S. Busquier & Á. A. Magreñán, 2013. "Reducing Chaos and Bifurcations in Newton‐Type Methods," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    5. Xiubin Xu & Yuan Xiao & Tao Liu, 2012. "Semilocal Convergence Analysis for Inexact Newton Method under Weak Condition," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    6. Rongfei Lin & Yueqing Zhao & Zdeněk Šmarda & Yasir Khan & Qingbiao Wu, 2013. "Newton-Kantorovich and Smale Uniform Type Convergence Theorem for a Deformed Newton Method in Banach Spaces," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-8, December.
    7. Jisheng Kou & Shuyu Sun & Bo Yu, 2011. "Multiscale Time-Splitting Strategy for Multiscale Multiphysics Processes of Two-Phase Flow in Fractured Media," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-24, March.
    8. Rongfei Lin & Yueqing Zhao & Zdeněk Šmarda & Yasir Khan & Qingbiao Wu, 2013. "Newton‐Kantorovich and Smale Uniform Type Convergence Theorem for a Deformed Newton Method in Banach Spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    9. S. Amat & S. Busquier & Á. A. Magreñán, 2013. "Reducing Chaos and Bifurcations in Newton-Type Methods," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-10, July.
    10. Xiubin Xu & Yuan Xiao & Tao Liu, 2012. "Semilocal Convergence Analysis for Inexact Newton Method under Weak Condition," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-13, August.
    11. Yonghui Ling & Xiubin Xu & Shaohua Yu, 2013. "Convergence Behavior for Newton-Steffensen’s Method under -Condition of Second Derivative," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-11, November.
    12. Yonghui Ling & Xiubin Xu & Shaohua Yu, 2013. "Convergence Behavior for Newton‐Steffensen’s Method under γ‐Condition of Second Derivative," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
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