GPGCD：计算一元多项式的近似GCD的迭代法
GPGCD: An iterative method for calculating approximate GCD of univariate polynomials
摘要：
给出了一种计算实系数或复系数一元多项式的近似最大公因数的迭代算法。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。将近似GCD问题转化为一个约束极小化问题，然后用变形牛顿法迭代求解，该方法是梯度投影法的推广.我们证明，在一些测试情况下，我们的算法计算近似GCD的扰动一样小的结构总最小范数（STLN）方法和UVGCD方法的基础上计算的方法，而我们的方法运行速度显着比他们的约高达30或10倍，分别与他们的实施相比。我们还表明，我们的算法适当地处理一些病态多项式，其中有一个小或大的首项系数的GCD。
英文摘要：
We present an iterative algorithm for calculating approximate greatest common divisor (GCD) of univariate polynomials with the real or 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. The problem of approximate GCD is transfered to a constrained minimization problem, then solved with the so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. We demonstrate that, in some test cases, our algorithm calculates approximate GCD with perturbations as small as those calculated by a method based on the structured total least norm (STLN) method and the UVGCD method, while our method runs significantly faster than theirs by approximately up to 30 or 10 times, respectively, compared with their implementation. We also show that our algorithm properly handles some ill-conditioned polynomials which have a GCD with small or large leading coefficient.
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