Fifth-Order Iterative Method for Solving Multiple Roots of the Highest Multiplicity of Nonlinear Equation

Liang, Juan; Li, Xiaowu; Wu, Zhanghua; Zhang, Mingsheng; Wang, Lin; Pan, Feng

doi:10.3390/a8030656

Cited by 2 publications

(3 citation statements)

References 27 publications

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“…Proof. We adopt the numerical analysis method which is equivalent to those in the literature [41,42]. Firstly, we deduce the expression of footpoint q.…”

Section: Convergence Analysis Of the H-h-h Methodsmentioning

confidence: 99%

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

Liang

Hou

et al. 2018

Mathematics

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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.

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“…Proof. We adopt the numerical analysis method which is equivalent to those in the literature [41,42]. Firstly, we deduce the expression of footpoint q.…”

Section: Convergence Analysis Of the H-h-h Methodsmentioning

confidence: 99%

“…Secondly, we deduce that the convergence order of the method defined by (2) or (3) is first order convergent. Our proof method absorbs the idea of [41,42]. Substituting (8) into (2), and simplifying, we get the relationship,…”

Section: Convergence Analysis Of the H-h-h Methodsmentioning

confidence: 99%

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

Liang

Hou

et al. 2018

Mathematics

Self Cite

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“…We use the method of the numerical analysis similar to those in the literature [24][25][26][27][28].…”

Section: Convergence Analysismentioning

confidence: 99%

A Geometric Orthogonal Projection Strategy for Computing the Minimum Distance Between a Point and a Spatial Parametric Curve

Hou

et al. 2016

Algorithms

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Abstract:A new orthogonal projection method for computing the minimum distance between a point and a spatial parametric curve is presented. It consists of a geometric iteration which converges faster than the existing Newton's method, and it is insensitive to the choice of initial values. We prove that projecting a point onto a spatial parametric curve under the method is globally second-order convergence.

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Fifth-Order Iterative Method for Solving Multiple Roots of the Highest Multiplicity of Nonlinear Equation

Cited by 2 publications

References 27 publications

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

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

A Geometric Orthogonal Projection Strategy for Computing the Minimum Distance Between a Point and a Spatial Parametric Curve

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