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Open AccessArticle

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

by Juan Liang 1,2,†, Linke Hou 3,*,†, Xiaowu Li 4,*,†, Feng Pan 4,†, Taixia Cheng 5,† and Lin Wang 4,†
1
Data Science and Technology, North University of China, Taiyuan 030051, Shanxi, China
2
Department of Science, Taiyuan Institute of Technology, Taiyuan 030008, Shanxi, China
3
Center for Economic Research, Shandong University, Jinan 250100, Shandong, China
4
College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, Guizhou, China
5
Graduate School, Guizhou Minzu University, Guiyang 550025, Guizhou, China
*
Authors to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2018, 6(12), 306; https://doi.org/10.3390/math6120306
Received: 16 October 2018 / Revised: 25 November 2018 / Accepted: 28 November 2018 / Published: 5 December 2018
(This article belongs to the Special Issue Iterative Methods for Solving Nonlinear Equations and Systems)
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. View Full-Text
Keywords: point projection; intersection; parametric curve; n-dimensional Euclidean space; Newton’s second order method; fixed point theorem point projection; intersection; parametric curve; n-dimensional Euclidean space; Newton’s second order method; fixed point theorem
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Liang, J.; Hou, L.; Li, X.; Pan, F.; Cheng, T.; Wang, L. Hybrid Second Order Method for Orthogonal Projection onto Parametric Curve in n-Dimensional Euclidean Space. Mathematics 2018, 6, 306.

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