Designing a Tetration Library

[bo198214]

but for the hyperbolic case, you seem to be using a method I've never seen before. Embedded in your implementation is the implicit recurrence equation:

I described it here. The principle \( f^{\circ t} \circ f=f\circ f^{\circ t} \) with \( {f^{\circ t}}_1={f_1}^t \) is common in the regular iteration literature, however the formula is perhaps too dirty that I have seen it explicitely mentioned.

(1) Several methods are missing underscores (for example _repr_ should be __repr__)

You can read it in the sage package guidelines somewhere that you should use the sage specific _repr_ so that the user has the possibility to redefine __repr__

(2) Your inverse() method is empty, but I have implemented series_inverse() already, so we can use this.

Yes, the nice thing about the (hyp,par) iterate method is, that it is universally also applicable for exponent (-1). So the inverse function is just a wrapper for iterate(-1).

However perhaps in the future I will also include the direct formulas, for example to take the compositional root, you can easily derive an iterative formula solving \( f^{\circ n} = g \) for \( f \). Though the result should be the same as if you just do \( g^{\circ 1/n} \) vie the regular iteration. I also plan to implement the power \( f^{x} \) for arbitrary \( x \) and not just for integers as it is now.

Originally I wanted to use "^" via __and__ for the iteration, like I used "**" via __pow__ for the powers. Do you have an idea why this is not recognized? And also the question about how to put the hyperops package into sage that it is automatically recognized is still open for me.