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

Suppose we have a complex polynomial f ( z ) whose coefficients are inaccurate, and a prescribed complex number α such that f ( 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 l p -norm, weighted l p -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. [ABSTRACT FROM AUTHOR]