Approximate common divisors of polynomials and degree reduction for rational curves
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.
scientific article; zbMATH DE number 1286287