A unified approach to computing the nearest complex polynomial with a given zero

• This paper introduces a new synthetic block-wise norm and proves its dual norm which is very important here.
• This paper for the first time proposes a unified approach to compute the nearest complex polynomial with a given zero.
• This paper discusses a matrix-valued optimization problem that is very common in machine learning as an application of the introduced norm.

Suppose we have a complex polynomial f(z)f(z) whose coefficients are inaccurate, and a prescribed complex number α   such that f(α)≠0f(α)≠0. We study the problem of computing a complex polynomial f˜(z) such that f˜(α)=0 and the distance between f˜ and f  , i.e. ‖f˜−f‖, is minimal. Considering that previous works usually took the usual lplp-norm, weighted lplp-norm and block-wise norm as distance measures, we first introduce a new-defined synthetic norm that integrates all these norms. Then, we propose a unified approach to study the proposed problem and succeed in giving explicit expressions of the nearest polynomial. The effectiveness of our approach is illustrated by two examples, one of which shows an extension of finding the nearest complex polynomial with a zero in a given domain. Finally, as an application of the new-defined norm, we discuss a matrix-valued optimization problem that is very common in machine learning.