Zeration = inconsistant ?

[MphLee]

Example :

Solve

exp^[2]( 2 ln^[2](x) ) = 7

This is equivalent to

x^ln(x) = 7.

Take some real a,b >= exp(1).

a_0 = a^{ 1 / ln(b) }
b_0 = 7^{ ln(b) / ln(a) }

replacement rules :
---
a ' = a * 7^{ ln(b) }

b ' = a * b
---

a_1 = a ' ^{ 1/ln(b ') }
b_1 = 7^{ ln(b ') / ln(a ') }

repeat forever

lim n-> oo

a_n/b_n = x.

this gives x = exp( sqrt( ln(7) ) ) as it should.

Numerically we get x = 4.0348084730118923250275859453110072467762717139110...

Notice 1/x is also a solution.
If we take 0 < a,b < 1/exp(1) ( =exp(-1) ) we probably achieve that.

This numerical algorithm can probably be improved with adding some + operators at the right places ...

Still investigating.

Really interesting and ingenious but I'm a donkey with this kind of math. In other words, I dont see why you should use this sophisticated method when the equation can be solved easily by

Also the method can probably be extended nicely to all interpretations of hyperoperators.

How?You are talking about every sequence Hyperoperations sequence??

Can zeration inprove this algoritm ?

I have no idea