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

Juan Liang, Linke Hou, Xiaowu Li, Feng Pan, Taixia Cheng and Lin Wang
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Juan Liang: Data Science and Technology, North University of China, Taiyuan 030051, Shanxi, China
Linke Hou: Center for Economic Research, Shandong University, Jinan 250100, Shandong, China
Xiaowu Li: College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, Guizhou, China
Feng Pan: College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, Guizhou, China
Taixia Cheng: Graduate School, Guizhou Minzu University, Guiyang 550025, Guizhou, China
Lin Wang: College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025, Guizhou, China

Mathematics, 2018, vol. 6, issue 12, 1-23

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.

Keywords: point projection; intersection; parametric curve; n -dimensional Euclidean space; Newton’s second order method; fixed point theorem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2018
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