Hi.

I wanted to report to you some results I had trying out a new tetration method, or, well, actually a new twist on an old method. It's based on Kouznetsov's Cauchy integral method, only with a new and powerful method to solve the integral equation. It's actually just a new way of solving the integral equation in the Cauchy integral method.

What was the problem with Kouznetsov's method? While now it seems like it works for real bases greater than \( \eta \) and a lot of complex bases, it doesn't seem to work for the difficult challenge bases \( b = -1 \), \( b = 0.04 \), and \( b = e^{1/e} \) (Actually, at first I didn't think it worked for complex bases at all, but some more playing recently showed it to work with them with a slight modification (averaging successive iterations together, and using Kouznetsov's suggesting of updating the even and odd-index nodes separately), though still no dice with regards to base -1 (didn't try the other two), even taking the log caveat into account (see end here). Hence the need for this new method still remains.).

So for the goal of those "challenge bases", such as base -1, I've tried a more sophisticated approach to solving the Cauchy integral equation. Some googling on integral equations, specifically "nonlinear Fredholm of second kind" turned up a number of methods, some of which, such as Newton-Kantorovich and "Haar wavelets", were tried, but without much luck. But now, I think I have at last found something that could work.

It is HAM -- the Homotopy Analysis Method. From my experimentation, this method looks to be so good that it can not only tetrate complex bases, but perhaps even tetrate them all, including exotic bases such as \( -1 \), \( 0.04 \), and, of course... \( e^{-e} \), which have proven notoriously difficult to tetrate with other methods. The case \( e^{-e} \) is really interesting since it lies on the Shell-Thron border and has pseudo-period 2 at the principal fixed point, and as far as I can tell, there's hasn't yet been a good construction of the merged/bipolar superfunction (i.e. the tetrational, \( \mathrm{tet} \)) at this base. sheldonison mentioned some work toward this, though. Nonetheless, with the HAM, it looks to be possible to construct what would be that superfunction, and perhaps this might point the way towards tetrating it with sheldonison's merge method, or just providing a new, independent method, though it seems choosing the right initial guess is one of the tricky aspects here. I have managed, however, to successfully tetrate base \( -1 \).

Note that what I've got so far is experimental: there are a number of parameters in the HAM of which I am not yet sure how to set best, including the initial guess (which seems like it could use improvement), to optimize convergence, so right now is slower than Kouznetsov's original algorithm. But so far it seems like it should work for tetrating possibly all bases, and maybe even achieving the analytic continuation of the tetration to its other branches in the base-parameter (e.g. by whirling around the singularities at \( b = 1 \) and \( b = 0 \)), though I haven't tested that last part out yet. There is a caveat with regards to base -1 that requires some explanation (has to do with the multivaluedness of the complex logarithm), but that was easily taken care of. I'm also going to try tetrating base \( e^{-e} \) with it.

Are you interested? If so, I could post more posts detailing the method (I've already got a lot of it written up) as well as some PARI/GP code to play with, including some to tetrate base -1 (although it converges slowly -- I believe I need the proper values of the free parameters to make that work, but I made this code to play around with the method and get to know it better, and haven't yet seen anything about how to properly choose those parameters). I wonder how this method will stack up to other methods once it has been tuned.

I wanted to report to you some results I had trying out a new tetration method, or, well, actually a new twist on an old method. It's based on Kouznetsov's Cauchy integral method, only with a new and powerful method to solve the integral equation. It's actually just a new way of solving the integral equation in the Cauchy integral method.

What was the problem with Kouznetsov's method? While now it seems like it works for real bases greater than \( \eta \) and a lot of complex bases, it doesn't seem to work for the difficult challenge bases \( b = -1 \), \( b = 0.04 \), and \( b = e^{1/e} \) (Actually, at first I didn't think it worked for complex bases at all, but some more playing recently showed it to work with them with a slight modification (averaging successive iterations together, and using Kouznetsov's suggesting of updating the even and odd-index nodes separately), though still no dice with regards to base -1 (didn't try the other two), even taking the log caveat into account (see end here). Hence the need for this new method still remains.).

So for the goal of those "challenge bases", such as base -1, I've tried a more sophisticated approach to solving the Cauchy integral equation. Some googling on integral equations, specifically "nonlinear Fredholm of second kind" turned up a number of methods, some of which, such as Newton-Kantorovich and "Haar wavelets", were tried, but without much luck. But now, I think I have at last found something that could work.

It is HAM -- the Homotopy Analysis Method. From my experimentation, this method looks to be so good that it can not only tetrate complex bases, but perhaps even tetrate them all, including exotic bases such as \( -1 \), \( 0.04 \), and, of course... \( e^{-e} \), which have proven notoriously difficult to tetrate with other methods. The case \( e^{-e} \) is really interesting since it lies on the Shell-Thron border and has pseudo-period 2 at the principal fixed point, and as far as I can tell, there's hasn't yet been a good construction of the merged/bipolar superfunction (i.e. the tetrational, \( \mathrm{tet} \)) at this base. sheldonison mentioned some work toward this, though. Nonetheless, with the HAM, it looks to be possible to construct what would be that superfunction, and perhaps this might point the way towards tetrating it with sheldonison's merge method, or just providing a new, independent method, though it seems choosing the right initial guess is one of the tricky aspects here. I have managed, however, to successfully tetrate base \( -1 \).

Note that what I've got so far is experimental: there are a number of parameters in the HAM of which I am not yet sure how to set best, including the initial guess (which seems like it could use improvement), to optimize convergence, so right now is slower than Kouznetsov's original algorithm. But so far it seems like it should work for tetrating possibly all bases, and maybe even achieving the analytic continuation of the tetration to its other branches in the base-parameter (e.g. by whirling around the singularities at \( b = 1 \) and \( b = 0 \)), though I haven't tested that last part out yet. There is a caveat with regards to base -1 that requires some explanation (has to do with the multivaluedness of the complex logarithm), but that was easily taken care of. I'm also going to try tetrating base \( e^{-e} \) with it.

Are you interested? If so, I could post more posts detailing the method (I've already got a lot of it written up) as well as some PARI/GP code to play with, including some to tetrate base -1 (although it converges slowly -- I believe I need the proper values of the free parameters to make that work, but I made this code to play around with the method and get to know it better, and haven't yet seen anything about how to properly choose those parameters). I wonder how this method will stack up to other methods once it has been tuned.