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.

中文翻译：

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$ q $-完整序列的$ N $项的快速计算和应用

1977年，斯特拉森（Strassen）发明了著名的婴儿步/巨型步算法，该算法以$ \ sqrt {N} $的算术复杂度拟线性方式计算阶乘$ N！$。1988年，Chudnovsky兄弟将Strassen算法推广到以几乎相同的算术复杂度计算任何完整序列的第N $$项的计算。我们设计这些算法的$ q $类似物。我们首先将Strassen算法扩展到$ N $的$ q $阶乘的计算，然后将Chudnovskys算法扩展到任何$ q $完整序列的$ N $项的计算。两种算法都在$ \ sqrt {N} $中以算术复杂度准线性工作；令人惊讶的是，在完整的情况下，它们比类似物更简单。我们提供算术模型和位复杂度模型的详细成本分析。此外，我们描述了各种算法后果，