一种基于Pellet测试和牛顿迭代的近最优子区域算法用于复根隔离
A Near-Optimal Subdivision Algorithm for Complex Root Isolation based on the Pellet Test and Newton Iteration
AI总结：
本文提出一种基于Pellet测试和牛顿迭代的子区域算法，用于复根隔离，通过分析根的几何特性，实现近最优的子区域步骤，该算法在复杂度上优于传统方法，且不依赖分治技术。
摘要：
我们描述了一种用于隔离多项式$F\in\mathbb{C}[x]$复根的子区域算法。给定一个提供系数近似值的Oracle，以及一个包含仅简单根的任意复平面上的正方形$\mathcal{B}$，我们的算法返回互不重叠的隔离盘。我们的复杂度分析界定了系数近似精度、总迭代次数和整体位复杂度。进一步显示，算法的复杂度由输入正方形$\mathcal{B}$附近根的几何特性控制，即根的数量、绝对值和 pairwise 距离。子区域步骤的数量近似最优。对于基准问题，即隔离一个度数为$n$、整数系数位大小小于$τ$的多项式的所有根，我们的算法需要$\tilde O(n^3+n^2τ)$位操作，与Pan(2002)的记录界相当。这是首次使用子区域方法达到此类界，且不依赖Schönhage的分割圆技术。我们的算法使用Weyl(1924)的四叉树构造，结合Pellet定理与Graeffe迭代，推导出“软测试”以计数盘内的根数量。结合Schröder的修改牛顿算子与二分法，以Abbot(2006)的二次区间方法为灵感，实现对根簇的二次收敛。相对于分治算法，我们的算法较为简单，具有实际应用潜力。本文自包含：我们为所用的子程序提供了伪代码。
英文摘要：
We describe a subdivision algorithm for isolating the complex roots of a polynomial $F\in\mathbb{C}[x]$. Given an oracle that provides approximations of each of the coefficients of $F$ to any absolute error bound and given an arbitrary square $\mathcal{B}$ in the complex plane containing only simple roots of $F$, our algorithm returns disjoint isolating disks for the roots of $F$ in $\mathcal{B}$. Our complexity analysis bounds the absolute error to which the coefficients of $F$ have to be provided, the total number of iterations, and the overall bit complexity. It further shows that the complexity of our algorithm is controlled by the geometry of the roots in a near neighborhood of the input square $\mathcal{B}$, namely, the number of roots, their absolute values and pairwise distances. The number of subdivision steps is near-optimal. For the \emph{benchmark problem}, namely, to isolate all the roots of a polynomial of degree $n$ with integer coefficients of bit size less than $τ$, our algorithm needs $\tilde O(n^3+n^2τ)$ bit operations, which is comparable to the record bound of Pan (2002). It is the first time that such a bound has been achieved using subdivision methods, and independent of divide-and-conquer techniques such as Schönhage's splitting circle technique. Our algorithm uses the quadtree construction of Weyl (1924) with two key ingredients: using Pellet's Theorem (1881) combined with Graeffe iteration, we derive a "soft-test" to count the number of roots in a disk. Using Schröder's modified Newton operator combined with bisection, in a form inspired by the quadratic interval method from Abbot (2006), we achieve quadratic convergence towards root clusters. Relative to the divide-conquer algorithms, our algorithm is quite simple with the potential of being practical. This paper is self-contained: we provide pseudo-code for all subroutines used by our algorithm.
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