论文标题
计算和索引迭代指数的固定点的系统方法
A systematic approach to computing and indexing the fixed points of an iterated exponential
论文作者
论文摘要
本文介绍了一种系统计算和索引固定点的系统方法，用于固定的$ z $或等效地，$ t_2（w; z）= w-z^{z^w} $的根。根源是使用修改版本的定点迭代的修改版本计算的，并由整数三重态索引$ \ {n，m，p \} $，该版本将根与$ t_2 $的唯一分支相关联。提出了此命名约定，足以用$ \ Mathbb {z}^2 $枚举的$（n，m）$枚举该功能的所有根。但是，附近的分支可以具有多个根。这些情况由第三个参数$ p $确定。这项工作是用$ z $的理性或符号值完成的，实现了任意的精确算术。将根的精选为$ \ {10^{12}，10^{12}，p \} $带有$ | z | | \ leq 10^{12} $，用作测试用例。结果对于计算中使用的精度是准确的，通常在$ 30 $至100美元之间。 Mathematica Ver。 $ 12 $用于实施算法。
This paper describes a systematic method of numerically computing and indexing fixed points of $z^{z^w}$ for fixed $z$ or equivalently, the roots of $T_2(w;z)=w-z^{z^w}$. The roots are computed using a modified version of fixed-point iteration and indexed by integer triplets $\{n,m,p\}$ which associate a root to a unique branch of $T_2$. This naming convention is proposed sufficient to enumerate all roots of the function with $(n,m)$ enumerated by $\mathbb{Z}^2$. However, branches near the origin can have multiple roots. These cases are identified by the third parameter $p$. This work was done with rational or symbolic values of $z$ enabling arbitrary precision arithmetic. A selection of roots up to order $\{10^{12},10^{12},p\}$ with $|z|\leq 10^{12}$ was used as test cases. Results were accurate to the precision used in the computations, generally between $30$ and $100$ digits. Mathematica ver. $12$ was used to implement the algorithms.
