The Nearest Complex Polynomial with a Prescribed Zero 
Received:July 07, 2014 Revised:October 13, 2014 
Key Word:
nearest polynomial explicit expression zero dual norm

Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.61432003; 11171052; 61272371; 61328206; 11361005), the Research Programs of Gannan Normal University (Grant No.14zb21) and College of Mathematics and Computer Science. 

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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 $\\tff\$ 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 fourstep 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. 
Citation: 
DOI:10.3770/j.issn:20952651.2015.01.004 
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