给定零点的最近复系数多项式计算问题

The Nearest Complex Polynomial with a Prescribed Zero

中文关键词:  最近多项式  显式表达式  零点  对偶范数

英文关键词:nearest polynomial  explicit expression  zero  dual norm

中文摘要:

      满足某些给定性质的最近多项式在控制论和应用数学领域具有广泛的应用.给定一元多项式$f(z)$与零点$\alpha$,本文研究一元复系数多项式$\tilde f(z)$的计算问题,其中使$\tilde f(z)$ 满足$\tilde f(\alpha)=0$且离$f$的距离$\|\tilde f-f\|$最小.鉴于大部分现有工作对该问题的研究仅限于某些特定的多项式基底和向量范数,本文建立了一个基于一般多项式基底和向量范数的理论框架,并给出了计算$\tilde f(z)$的四个具体步骤.进一步,为获得$\tilde f(z)$的显式表达式,同时推广现有工作中惯用的$\ell_p$范数和混合范数,本文引入了两种新的向量范数,然后根据算法给出了$\tilde f(z)$ 在不同参数取值时的显式表达式.最后,数值例子验证了算法的有效性.

英文摘要:

      Nearest polynomial with given properties has many applications in control theory and applied mathematics. Given a complex univariate polynomial $f(z)$ and a zero $\alpha$, in this paper we explore the problem of computing a complex polynomial $\tilde{f}(z)$ such that $\tilde{f}(\alpha)=0$ and the distance $\|\tf-f\|$ is minimal. Considering most of the existing works focus on either certain polynomial basis or certain vector norm, we propose a common computation framework based on both general polynomial basis and general vector norm, and summarize the computing process into a four-step algorithm. Further, to find the explicit expression of $\tilde f(z)$, we focus on two specific norms which generalize the familiar $\ell_p$-norm and mixed norm studied in the existing works, and then compute $\tilde f(z)$ explicitly based on the proposed algorithm. We finally give a numerical example to show the effectiveness of our method.