Certified simultaneous isotopic approximation of algebraic curves via subdivision

IF 0.6 4区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Journal of Symbolic Computation Pub Date : 2025-02-26 DOI:10.1016/j.jsc.2025.102435

Michael Burr, Michael Byrd

引用次数: 0

Abstract

We present a certified algorithm based on subdivision for computing an isotopic approximation to any number of algebraic curves in the plane. Our algorithm is based on the certified curve approximation algorithm of Plantinga and Vegter. The main challenge in this algorithm is to correctly and efficiently identify and isolate all intersections between the curves. To overcome this challenge, we introduce a new and simple test that guarantees the global correctness of our output. A main step in our algorithm for approximating any number of curves is to correctly approximate a pair of curves. In addition to developing the details of this special case, we provide complexity analyses for both the number of steps and the bit-complexity of this algorithm using both worst-case bounds as well as those based on continuous amortization and condition numbers.

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通过细分的代数曲线的认证同时同位素近似

我们提出了一种基于细分的证明算法，用于计算平面上任意数量代数曲线的同位素近似。该算法基于Plantinga和vetter的认证曲线近似算法。该算法的主要挑战是正确有效地识别和隔离曲线之间的所有交点。为了克服这一挑战，我们引入了一个新的简单测试，以保证输出的全局正确性。在我们的算法中近似任意数量的曲线的一个主要步骤是正确地近似一对曲线。除了开发这种特殊情况的细节外，我们还使用最坏情况边界以及基于连续摊销和条件数的边界，对该算法的步骤数和位复杂度进行了复杂性分析。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Symbolic Computation 工程技术-计算机：理论方法

CiteScore

2.10

自引率

14.30%

发文量

审稿时长

142 days

期刊介绍： An international journal, the Journal of Symbolic Computation, founded by Bruno Buchberger in 1985, is directed to mathematicians and computer scientists who have a particular interest in symbolic computation. The journal provides a forum for research in the algorithmic treatment of all types of symbolic objects: objects in formal languages (terms, formulas, programs); algebraic objects (elements in basic number domains, polynomials, residue classes, etc.); and geometrical objects. It is the explicit goal of the journal to promote the integration of symbolic computation by establishing one common avenue of communication for researchers working in the different subareas. It is also important that the algorithmic achievements of these areas should be made available to the human problem-solver in integrated software systems for symbolic computation. To help this integration, the journal publishes invited tutorial surveys as well as Applications Letters and System Descriptions.