GPGCD，一种计算复系数一元多项式近似GCD的迭代法
GPGCD, an Iterative Method for Calculating Approximate GCD of Univariate Polynomials, with the Complex Coefficients
摘要：
本文将计算一元多项式的近似最大公因数的迭代法GPGCD推广到复系数多项式。对于给定的一对多项式和一个次数，我们的算法找到一对多项式，它具有给定次数的GCD，并且其系数与原始输入中的系数相比被扰动，使得扰动尽可能小，同时具有GCD。在GPGCD方法中，近似GCD问题转化为一个约束极小化问题，然后用一个所谓的变形牛顿法求解，该方法是梯度投影法的推广，通过迭代搜索解。而我们原来的方法是设计的多项式与实系数，我们将其扩展到接受多项式与复系数在本文中。
英文摘要：
We present an extension of our GPGCD method, an iterative method for calculating approximate greatest common divisor (GCD) of univariate polynomials, to polynomials with the complex coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. In our GPGCD method, the problem of approximate GCD is transfered to a constrained minimization problem, then solved with a so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. While our original method is designed for polynomials with the real coefficients, we extend it to accept polynomials with the complex coefficients in this paper.
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