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