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.

翻译：本文展示了用于计算多元最大共同比值的正规化理论和趋同分析,同时详细描述黑盒型算法。常规GCD计算中的错误根源由其几何特征确定,其中多球体形成不同的多球体,交织在一个分层结构中。有了适当的正规化,数字的GCD被证明是非常可靠的。最重要的是,数字的GCD解决了使用浮动点算术来准确找到GCD的问题,即使数据被绕过。敏感度测量、每个计算阶段的错误界限和总体趋同都得到了严格确定。选定的测试实例的计算结果显示,算法和软件看起来是稳健的和准确的。