Approximate common divisors of polynomials and degree reduction for rational curves
type: Item
instance of: scholarly article
title: Approximate common divisors of polynomials and degree reduction for rational curves
zbMATH Open document ID: 0923.65005
author: Jianzhong Sun
author: Falai Chen
author: Yongming Qu
published in: Applied Mathematics. Series B (English Edition)
publication date: 1999-10-28T00:00:00Z
review text: A new algorithm is presented in this paper to solve the problem of finding approximated common divisors of a set of polynomials \(\vec f(t)= \{f_1(t),\dots, f_m(t)\}\) of degree \(n\) on the interval \([a,b]\). The procedure consists in constructing a set of perturbation polynomials \(\vec\varepsilon= \{\varepsilon_1,\dots,\varepsilon_m\}\) such that \(\vec f+\vec\varepsilon\) have a common divisor and \(\vec\varepsilon(t)\) is minimized under certain norm. The set of polynomials \(\vec f+\vec\varepsilon\) interpolate \(\vec f\) at the endpoints of the interval. Hence, this method when combined with a subdivision technique yields \(C^r\) continuous piecewise approximation for any rational curve. Some additional details and results are given below. The following representation (B-B form) is used for all the polynomials: \(f_i(t)= \sum^n_{j=0} f_{ij}B^n_j(t)\), where \(B^n_j(t)= \begin{pmatrix} n\\ j\end{pmatrix}\left({t-a\over b-a}\right)^j\left({b-t\over b-a}\right)^{n-j}\), \(j= 0,\dots, n\), are Bernstein basis functions of degree \(n\) on the interval \([a,b]\). The set of perturbation polynomials are written in B-B form using the notation \(\varepsilon_{ij}\) for scalars. The optimization problem to be solved is \(\min z= \sum^m_{i= 1}\sum^n_{j= 0}\varepsilon^2_{ij}\), subject to the interpolation constraints \(\vec\varepsilon^{(k)}(a)= \vec\varepsilon^{(k)}(b)= 0\), \(k= 0,1,\dots,\mu\) (\(\mu\leq [(n- 2)/2]\)). The following results are proved. Given an arbitrary constant \(\alpha\neq 0, 1\) there exists a perturbation polynomial set \(\vec\varepsilon(t)\), which is a solution to the above optimization problem, such that \(\vec f+ \vec\varepsilon\) has common divisor \(t-\alpha\). The polynomial \(t-\alpha\) is the best approximate linear common divisor of \(\vec f\). The final section is devoted to the degree reduction of rational curves. The paper contains two numerical examples.
Mathematics Subject Classification ID: 65D20
Mathematics Subject Classification ID: 65D05
Mathematics Subject Classification ID: 65D17
Mathematics Subject Classification ID: 12Y05
zbMATH DE Number: 1286287
zbMATH Keywords: approximate common divisors
zbMATH Keywords: rational curves
zbMATH Keywords: polynomials
zbMATH Keywords: subdivision
zbMATH Keywords: Bernstein basis functions
zbMATH Keywords: perturbation polynomials
zbMATH Keywords: interpolation
zbMATH Keywords: degree reduction
zbMATH Keywords: numerical examples
reviewed by: J. Illán
MaRDI profile type: Publication
full work available at URL: s11766-998-0056-9
OpenAlex ID: W2054334737
cites work: Best linear common divisors for approximate degree reduction
cites work: Improperly parametrized rational curves
cites work: A survey of curve and surface methods in CAGD
DOI: 10.1007/S11766-998-0056-9
description: scientific article; zbMATH DE number 1286287