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This paper describes a systematic method of numerically computing andindexing 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 offixed-point iteration and indexed by integer triplets $\{n,m,p\}$ whichassociate a root to a unique branch of $T_2$. This naming convention isproposed sufficient to enumerate all roots of the function with $(n,m)$enumerated by $\mathbb{Z}^2$. However, branches near the origin can havemultiple roots. These cases are identified by the third parameter $p$. Thiswork was done with rational or symbolic values of $z$ enabling arbitraryprecision 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 theprecision used in the computations, generally between $30$ and $100$ digits.Mathematica ver. $12$ was used to implement the algorithms.

A systematic approach to computing and indexing the fixed points of an iterated exponential