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Hybrid Second Order Method for Orthogonal Projection onto Parametric Curve in n -Dimensional Euclidean Space

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  • Juan Liang

    (Data Science and Technology, North University of China, Taiyuan 030051, Shanxi, China
    Department of Science, Taiyuan Institute of Technology, Taiyuan 030008, Shanxi, China
    These authors contributed equally to this work.)

  • Linke Hou

    (Center for Economic Research, Shandong University, Jinan 250100, Shandong, China
    These authors contributed equally to this work.)

  • Xiaowu Li

    (College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, Guizhou, China
    These authors contributed equally to this work.)

  • Feng Pan

    (College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, Guizhou, China
    These authors contributed equally to this work.)

  • Taixia Cheng

    (Graduate School, Guizhou Minzu University, Guiyang 550025, Guizhou, China
    These authors contributed equally to this work.)

  • Lin Wang

    (College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, Guizhou, China
    These authors contributed equally to this work.)

Abstract

Orthogonal projection a point onto a parametric curve, three classic first order algorithms have been presented by Hartmann (1999), Hoschek, et al. (1993) and Hu, et al. (2000) (hereafter, H-H-H method). In this research, we give a proof of the approach’s first order convergence and its non-dependence on the initial value. For some special cases of divergence for the H-H-H method, we combine it with Newton’s second order method (hereafter, Newton’s method) to create the hybrid second order method for orthogonal projection onto parametric curve in an n -dimensional Euclidean space (hereafter, our method). Our method essentially utilizes hybrid iteration, so it converges faster than current methods with a second order convergence and remains independent from the initial value. We provide some numerical examples to confirm robustness and high efficiency of the method.

Suggested Citation

  • Juan Liang & Linke Hou & Xiaowu Li & Feng Pan & Taixia Cheng & Lin Wang, 2018. "Hybrid Second Order Method for Orthogonal Projection onto Parametric Curve in n -Dimensional Euclidean Space," Mathematics, MDPI, vol. 6(12), pages 1-23, December.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:12:p:306-:d:188217
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    References listed on IDEAS

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    1. E. Polak & J. O. Royset & R. S. Womersley, 2003. "Algorithms with Adaptive Smoothing for Finite Minimax Problems," Journal of Optimization Theory and Applications, Springer, vol. 119(3), pages 459-484, December.
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