一元多项式的数值最大公因数
The numerical greatest common divisor of univariate polynomials
摘要：
本文给出了多项式最大公因数数值计算的正则化理论和收敛性分析，并详细描述了一个黑箱型算法。传统GCD计算中的不适定性的根源是由其几何结构确定的，其中多项式形成在分层结构中纠缠的微分流形。通过适当的正则化，证明了数值GCD是强适定的。最重要的是，数值GCD解决了使用浮点运算精确地找到GCD的问题，即使数据被扰动。一个灵敏度的测量，误差界在每个计算阶段，和整体收敛严格建立。通过算例计算表明，该算法和软件具有较好的鲁棒性和准确性。
英文摘要：
This paper presents a regularization theory for numerical computation of polynomial greatest common divisors and a convergence analysis, along with a detailed description of a blackbox-type algorithm. The root of the ill-posedness in conventional GCD computation is identified by its geometry where polynomials form differentiable manifolds entangled in a stratification structure. With a proper regularization, the numerical GCD is proved to be strongly well-posed. Most importantly, the numerical GCD solves the problem of finding the GCD accurately using floating point arithmetic even if the data are perturbed. A sensitivity measurement, error bounds at each computing stage, and the overall convergence are established rigorously. The computing results of selected test examples show that the algorithm and software appear to be robust and accurate.
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