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.
 Author Name Affiliation Wenyu HU College of Mathematics and Computer Science, Gannan Normal University, Jiangxi 341000, P. R. China Zhongxuan LUO School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China School of Software, Dalian University of Technology, Liaoning 116620, P. R. China
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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.