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arXiv · 2501.05182

An $O(n\log^2n)$ Algorithm for Computing Hankel Determinants up to Order $n$

Feihu Liu·Guoce Xin·Zihao Zhang
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Abstract

Given the rational power series $h(x) = \sum_{i \geq 0} h_i x^i \in \mathbb{C}[[x]]$, the Hankel determinant of order $n$ is defined as $H_n(h(x)) := \det (h_{i+j})_{1 \leq i,j \leq n}$. We explore the relationship between the Hankel continued fraction and the generalized Sturm sequence. This connection inspires the development of a novel algorithm for computing the Hankel determinants $\{H_i(h(x))\}_{i=0}^{n-1}$ using $O(n \log^2 n)$ arithmetic operations. We also explore the connection between the generalized Sturm sequences and the signature of Hankel matrices.

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Feihu Liu, Guoce Xin, Zihao Zhang. 2025-01-09. An $O(n\log^2n)$ Algorithm for Computing Hankel Determinants up to Order $n$. https://arxiv.org/abs/2501.05182

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