通过Padé逼近和Cauchy插值实现有理重构与XGCD的精细化复杂度界

中文摘要 AI 辅助

在单变量多项式计算中，两个基本且相关的问题是XGCD（扩展最大公因式）和有理重构，经典上通过半GCD算法以拟线性复杂度求解。这些问题在代数计算中有多种应用，并与线性递推序列、结构矩阵和连分数有紧密联系。本文首先给出了一系列算法归约，表明有理重构和XGCD可以通过模一个自由选择的多项式$M(x)$的关系基计算来求解。特别地，恢复了民间思想，即Padé逼近基（即$M(x) = x^d$）可用于执行拟线性有理重构或XGCD，将Berlekamp-Massey算法与扩展欧几里得算法之间的已知联系扩展到快速算法。这些归约的一个亮点是，除了逼近，还可以依赖Cauchy插值（即$M(x)$在选定点处为零）。在第二部分，本文描述了用于逼近和插值的分治算法，并给出了复杂度分析，显示了主导项前的显式前导常数。对于插值，最佳前导常数通过一种存储由求值表示的多项式的变体获得，并利用快速外推以避免重复转换到单项式基；这需要特殊点，如几何或算术级数中的点，或当基域允许时的FFT点。将分析与归约结合，得到了我们所知的有理重构和XGCD的最佳复杂度界。也许令人惊讶的是，即使是Padé逼近或类似Berlekamp-Massey的计算（其本质上涉及$M(x) = x^d$），也通过归约到在精心选择的点上的Cauchy插值而加速。

英文摘要

When computing with univariate polynomials, two fundamental and related problems are the XGCD and rational reconstruction, classically solved in quasi-linear complexity using the half-gcd algorithm. These problems have various applications in algebraic computations and bear strong connections to linearly recurrent sequences, structured matrices, and continued fractions. This article first gives a collection of algorithmic reductions, showing that rational reconstruction and XGCD can be solved via the computation of bases of relations modulo a freely-chosen polynomial $M(x)$. In particular, one recovers the folklore idea that bases of Padé approximants (i.e., $M(x) = x^d$) can be used to perform quasi-linear rational reconstruction or XGCD, extending to fast algorithms the well-known link between the Berlekamp-Massey algorithm and the extended Euclidean algorithm. One highlight of these reductions is that, instead of approximants, one may rely on Cauchy interpolants (i.e., $M(x)$ vanishes at chosen points). In a second part, this article describes divide-and-conquer algorithms for approximants and interpolants along with complexity analyses showing an explicit leading constant in front of the dominant term. For interpolants, the best leading constant is obtained through a variant that stores polynomials represented by evaluations, and exploits fast extrapolation in order to avoid repeated conversions to the monomial basis; this requires special points, in geometric or arithmetic progression, or FFT points when the base field allows them. Combining the analyses with the reductions leads to the best complexity bounds we are aware of for rational reconstruction and XGCD. Perhaps surprisingly, even Padé approximants or Berlekamp-Massey-like computations, which intrinsically involve $M(x) = x^d$, are accelerated by reducing them to Cauchy interpolation at well-chosen points.

发表机构

• Sorbonne Université, CNRS, LIP6（索邦大学，法国国家科学研究中心，LIP6）

机构由 AI 辅助整理，请以论文原文为准。