Fast Computation of the $N$-th Term of a $q$-Holonomic Sequence and Applications

Abstract : In 1977, Strassen invented a famous baby-step/giant-step algorithm that computes the factorial $N!$ in arithmetic complexity quasi-linear in $\sqrt{N}$. In 1988, the Chudnovsky brothers generalized Strassen’s algorithm to the computation of the $N$-th term of any holonomic sequence in essentially the same arithmetic complexity. We design $q$-analogues of these algorithms. We first extend Strassen’s algorithm to the computation of the $q$-factorial of $N$, then Chudnovskys' algorithm to the computation of the $N$-th term of any $q$-holonomic sequence. Both algorithms work in arithmetic complexity quasi-linear in $\sqrt{N}$; surprisingly, they are simpler than their analogues in the holonomic case. We provide a detailed cost analysis, in both arithmetic and bit complexity models. Moreover, we describe various algorithmic consequences, including the acceleration of polynomial and rational solving of linear $q$-differential equations, and the fast evaluation of large classes of polynomials, including a family recently considered by Nogneng and Schost.
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https://hal.inria.fr/hal-03084680
Contributor : Alin Bostan <>
Submitted on : Monday, December 21, 2020 - 12:04:30 PM
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• HAL Id : hal-03084680, version 1

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Alin Bostan, Sergey Yurkevich. Fast Computation of the $N$-th Term of a $q$-Holonomic Sequence and Applications. 2020. ⟨hal-03084680⟩

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