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.
Categories: math.NA cs.NA math.CV