严格的R. P. Brent的二元Euclidean algorithm模型
A rigorous version of R. P. Brent's model for the binary Euclidean algorithm
摘要：
二进制欧几里德算法是用于计算最大公因数的经典欧几里德算法的修改，其避免了普通整数除法，仅有利于除以2的幂。二进制欧几里德算法应用于有界大小的整数对时所需步骤数的期望值首先由R。P. Brent在1976年通过一个启发式模型把算法看作一个随机动力系统。通过对相关Ruelle转移算子的期望值的数值研究，Brent得到了该算法在处理大小受大整数限制的奇数对时所需平均步数的渐近表达式. 1998年B. Vallée通过一个归纳方案修改了Brent的模型，以严格证明该算法执行的平均步骤数的渐近公式;然而，这个结果与Brent的算法的关系仍然是不确定的。本文建立了Brent转移算子的先验性质，直接证明了它具有谱隙并保持唯一的连续密度。这个密度是全纯扩展到复杂的右半平面，并有一个对数奇点在零。将这些结果与经典解析数论的方法相结合，证明了三个关于期望步数的公式的正确性，解决了D. E. Knuth在《计算机程序设计的艺术》一书中。
英文摘要：
The binary Euclidean algorithm is a modification of the classical Euclidean algorithm for computation of greatest common divisors which avoids ordinary integer division in favour of division by powers of two only. The expectation of the number of steps taken by the binary Euclidean algorithm when applied to pairs of integers of bounded size was first investigated by R. P. Brent in 1976 via a heuristic model of the algorithm as a random dynamical system. Based on numerical investigations of the expectation of the associated Ruelle transfer operator, Brent obtained a conjectural asymptotic expression for the mean number of steps performed by the algorithm when processing pairs of odd integers whose size is bounded by a large integer. In 1998 B. Vallée modified Brent's model via an induction scheme to rigorously prove an asymptotic formula for the average number of steps performed by the algorithm; however, the relationship of this result with Brent's heuristics remains conjectural. In this article we establish previously conjectural properties of Brent's transfer operator, showing directly that it possesses a spectral gap and preserves a unique continuous density. This density is shown to extend holomorphically to the complex right half-plane and to have a logarithmic singularity at zero. By combining these results with methods from classical analytic number theory we prove the correctness of three conjectured formulae for the expected number of steps, resolving several open questions promoted by D. E. Knuth in The Art of Computer Programming.
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