给定零点的最近复系数多项式计算问题

Nearest polynomial with given properties has many applications in control theory and applied mathematics. Given a complex univariate polynomial f(z) and a zero α, in this paper we explore the problem of computing a complex polynomial f(z) such that f(α) = 0 and the distance ∥f-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 f(z), we focus on two specific norms which generalize the familiar lp-norm and mixed norm studied in the existing works, and then compute f(z) explicitly based on the proposed algorithm. We finally give a numerical example to show the effectiveness of our method.

The Nearest Complex Polynomial with a Prescribed Zero

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.