SqFreeEVAL：一个（几乎）最优的实根隔离算法
SqFreeEVAL: An (almost) optimal real-root isolation algorithm
摘要：
设f是一元实系数多项式，f在R[X]中.基于代数技术的细分算法（例如，Sturm或Descartes方法）被广泛用于隔离给定区间内f的实根。在本文中，我们考虑一个简单的细分算法，其基元是纯数值的（例如，功能评估）。该算法的复杂度是自适应的，因为该算法基于本地数据进行决策。自适应算法（特别是自适应算法）的复杂性分析是计算机科学面临的新挑战。在本文中，我们计算的尺寸的细分树的SqFreeEVAL算法。SqFreeEVAL算法是一种基于评估的数值算法，在多个社区中广为人知。该算法本身很简单，但先前计算其复杂性的尝试已被证明是相当技术性的，并产生了次优的结果。我们的主要结果是一个简单的O（d（L+ln d））上界的细分树的大小为SqFreeEVAL算法的基准问题隔离的所有实根的整数多项式f的度d，其系数可以写在最多L位。我们的证明使用了两种基于摊销的技术：首先，我们使用标准马勒-达文波特根界的代数摊销技术来解释d和L的积分。其次，我们使用一个连续摊销技术的基础上的积分来限制的大小的细分树。本文是第一次使用新的分析技术的连续摊销，以获得最先进的复杂性界限。
英文摘要：
Let f be a univariate polynomial with real coefficients, f in R[X]. Subdivision algorithms based on algebraic techniques (e.g., Sturm or Descartes methods) are widely used for isolating the real roots of f in a given interval. In this paper, we consider a simple subdivision algorithm whose primitives are purely numerical (e.g., function evaluation). The complexity of this algorithm is adaptive because the algorithm makes decisions based on local data. The complexity analysis of adaptive algorithms (and this algorithm in particular) is a new challenge for computer science. In this paper, we compute the size of the subdivision tree for the SqFreeEVAL algorithm. The SqFreeEVAL algorithm is an evaluation-based numerical algorithm which is well-known in several communities. The algorithm itself is simple, but prior attempts to compute its complexity have proven to be quite technical and have yielded sub-optimal results. Our main result is a simple O(d(L+ln d)) bound on the size of the subdivision tree for the SqFreeEVAL algorithm on the benchmark problem of isolating all real roots of an integer polynomial f of degree d and whose coefficients can be written with at most L bits. Our proof uses two amortization-based techniques: First, we use the algebraic amortization technique of the standard Mahler-Davenport root bounds to interpret the integral in terms of d and L. Second, we use a continuous amortization technique based on an integral to bound the size of the subdivision tree. This paper is the first to use the novel analysis technique of continuous amortization to derive state of the art complexity bounds.
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