论文标题
认证的牛顿方案，用于评估低生物theta功能
Certified Newton schemes for the evaluation of low-genus theta functions
论文作者
论文摘要
可以使用基于Borchardt序列的Newton方案在必需的精度下，在准线性时间的任何给定点上，可以在任何给定的点评估低属的theta函数和theta常数。本文我们的目标是提供必要的工具，以可证明正确的方式实施这些算法。特别地，我们在第1属和第2属的theta常数中获得了均匀和明确的收敛性，而theta在第1属中的功能：相关的牛顿方案将分别从近似值到N = 60、300和1600的精度的N位，以及所有可相当可减少的参数。我们还描述了一种均匀的准线性时间算法，以评估Siegel基本结构域上的2 theta常数。我们的主要工具是将Borchardt表示为多元分析功能的详细研究。
Theta functions and theta constants in low genus, especially genus 1 and 2, can be evaluated at any given point in quasi-linear time in the required precision using Newton schemes based on Borchardt sequences. Our goal in this paper is to provide the necessary tools to implement these algorithms in a provably correct way. In particular, we obtain uniform and explicit convergence results in the case of theta constants in genus 1 and 2, and theta functions in genus 1: the associated Newton schemes will converge starting from approximations to N bits of precision for N=60, 300, and 1600 respectively, for all suitably reduced arguments. We also describe a uniform quasi-linear time algorithm to evaluate genus 2 theta constants on the Siegel fundamental domain. Our main tool is a detailed study of Borchardt means as multivariate analytic functions.
