科研之友
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摘要
Suppose we have a complex polynomial f (z) whose coefficients are inaccurate, and a prescribed complex number alpha such that f (alpha) not equal 0. We study the problem of computing a complex polynomial (f) over tilde (z) such that (f) over tilde (alpha) = 0 and the distance between (f) over tilde and f, i.e. parallel to(f) over tilde - f parallel to, 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.
