The Nearest Complex Polynomial with a Prescribed Zero

Abstract:

      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.