求解具有一个自由度的多项式方程组算法的条件数分析
A condition number analysis of an algorithm for solving a system of polynomial equations with one degree of freedom
AI总结：
提出一种基于牛顿法和细分的算法，用于求解n+1个变量的n个实多项式方程组，定义条件数衡量与退化问题的距离，并证明运行时间受条件数、n和多项式次数限制。
摘要：
本文考虑求解$n+1$个变量中的$n$个实多项式方程组的问题。我们提出了一种基于牛顿法和细分的算法来解决这个问题。我们的算法仅适用于非退化情况，此时解是一维曲线。我们的第一个主要贡献是定义了一个条件数，用于测量与退化问题的倒数距离，该条件数可以区分该问题的良态和病态实例。（在我们的框架中，退化问题将是无限病态的。）我们的第二个贡献，也是我们算法的主要新颖之处，是分析表明其运行时间受问题实例的条件数以及$n$和多项式次数的限制。
英文摘要：
This article considers the problem of solving a system of $n$ real polynomial equations in $n+1$ variables. We propose an algorithm based on Newton's method and subdivision for this problem. Our algorithm is intended only for nondegenerate cases, in which case the solution is a 1-dimensional curve. Our first main contribution is a definition of a condition number measuring reciprocal distance to degeneracy that can distinguish poor and well conditioned instances of this problem. (Degenerate problems would be infinitely ill conditioned in our framework.) Our second contribution, which is the main novelty of our algorithm, is an analysis showing that its running time is bounded in terms of the condition number of the problem instance as well as $n$ and the polynomial degrees.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
