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Green Eggs and HAM: Tetration for ALL bases, real and complex, now possible?
#1
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 and a lot of complex bases, it doesn't seem to work for the difficult challenge bases , , and (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 , , and, of course... , which have proven notoriously difficult to tetrate with other methods. The case 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, ) 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 .

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 and ), 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 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.
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#2
(11/09/2013, 07:36 AM)mike3 Wrote: (...)
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.

Well, this all sounds very good. And as far I have seen this comes from someone who has provided always very reliable contributions here - so I'm much interested.
However, unfortunately... , I was never able to understand the work of Dmitri and so I also expect, I'll be off of your method - but I never mind: if we finally get some working method: how fine would that be! Perhaps, if you get this working and useful, we'll also find someone who could explain the mathematics for the intuition and for the lower degree math as well... :-)

Gottfried
Gottfried Helms, Kassel
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#3
(11/09/2013, 11:27 PM)Gottfried Wrote: Well, this all sounds very good. And as far I have seen this comes from someone who has provided always very reliable contributions here - so I'm much interested.
However, unfortunately... , I was never able to understand the work of Dmitri and so I also expect, I'll be off of your method - but I never mind: if we finally get some working method: how fine would that be! Perhaps, if you get this working and useful, we'll also find someone who could explain the mathematics for the intuition and for the lower degree math as well... :-)

Gottfried

So do you want me to post the description? I'm also curious: where do you have trouble with regards to Kouznetsov's method?
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#4
(11/09/2013, 07:36 AM)mike3 Wrote: 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 and a lot of complex bases, it doesn't seem to work for the difficult challenge bases , , and .....
I'm on my cellphone... not computer. This method sounds very exciting! You should publish it. Biggest problem with Kouznetsov's method is finite rectangle in imag (z) and discreet sampling. Perhaps an infinite rectangle ?Riemann mapping? to a unit circle? Probably not the approach you're thinking about ...

Anyway, I would definitely be interested in details on your new ideas, and look forward to subsequent posts. Does it work for real bases less than ? Kouznetsov's method relies on limiting behavior at , whereas these bases are periodic in .
- Sheldon

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#5
(11/10/2013, 02:45 AM)mike3 Wrote: So do you want me to post the description? I'm also curious: where do you have trouble with regards to Kouznetsov's method?
Well, surely to see the description were really nice (understanding & being-able-to-applicate-it even nicer :-) ) but if there is in fact something in it I'd propose to think about making a toolbox of the procedures and make a contract with Wolfram(mathematica) matlab and so on... - and only after this to publish the details.

The other aspect, what trouble I have with understanding the method: it was about 35 years ago that in some boring days of holiday I went to a library and found books about calculus/integration - written much better than those in my calculus-courses in german in my then college-times, such that I nearly became familiar with it. However, after being back home I became weak with this again and up to today I'm nearly illiterate with integration. Then the article even emphazises "Cauchy integration" and "contour integration" - stepping to "Riemann mapping" - and reading that text is then like trying to walk & balance on the pieces of ice in the arctic water... no reliable ground, no redundancy, - so even if I thought I might have got something correct I did not know whether this was true and meaningful to proceed. So I gave up with that text (I tried to step into it again a couple of times but with not much progress so far)...


Often it is only to understand some key-idea of a concept to be able to metabolize it completely, but that didn't happen so far with the above indicated concepts.

Gottfried
Gottfried Helms, Kassel
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#6
(11/10/2013, 07:38 PM)Gottfried Wrote:
(11/10/2013, 02:45 AM)mike3 Wrote: So do you want me to post the description? I'm also curious: where do you have trouble with regards to Kouznetsov's method?
Well, surely to see the description were really nice (understanding & being-able-to-applicate-it even nicer :-) ) but if there is in fact something in it I'd propose to think about making a toolbox of the procedures and make a contract with Wolfram(mathematica) matlab and so on... - and only after this to publish the details.

The other aspect, what trouble I have with understanding the method: it was about 35 years ago that in some boring days of holiday I went to a library and found books about calculus/integration - written much better than those in my calculus-courses in german in my then college-times, such that I nearly became familiar with it. However, after being back home I became weak with this again and up to today I'm nearly illiterate with integration. Then the article even emphazises "Cauchy integration" and "contour integration" - stepping to "Riemann mapping" - and reading that text is then like trying to walk & balance on the pieces of ice in the arctic water... no reliable ground, no redundancy, - so even if I thought I might have got something correct I did not know whether this was true and meaningful to proceed. So I gave up with that text (I tried to step into it again a couple of times but with not much progress so far)...


Often it is only to understand some key-idea of a concept to be able to metabolize it completely, but that didn't happen so far with the above indicated concepts.

Gottfried

You mention about making a "toolbox" for Wolfram Mathematica. Unfortunately, there's a few problems:

1. I'm not sure if the method is efficient enough to make tetration as rapidly computable as would be needed for such a program (right now (without tweaks) it's still not as fast as sheldonison's Kneser method and even that isn't fast enough,

2. I don't have Wolfram Mathematica myself (I can't afford it), so am not familiar with it/could not program anything for it (beyond what you get using Wolfram Alpha, of course). Ditto for Matlab -- can't afford that either and so have never used it. If you sat me down in front of either of these systems, I wouldn't really be able to do much.

3. I'm not sure what practical uses would exist for continuous tetration, which would make this worthwhile for such programs.

4. if you mean make code for the HAM (since it can be used to solve more than just tetration) in general, written in the languages of those programs, that might be interesting, but HAM code might already exist out there and I'm not sure what advantage any I might make would have.

As for the second point, sorry to hear about your situation with regards to knowledge of integration theory. Have you tried to go back and start from the beginning, as opposed to just jumping into the advanced stuff first?
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#7
(11/09/2013, 07:36 AM)mike3 Wrote: .... The case 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, ) 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 .
I'm also interested in any results for , with pseudo period 2, as per our earlier discussion on this forum. I haven't gotten any further with that base, though I believe it has a solution possible due to results I've gotten for a base with pseudo period=5, via a cumbersome indirect method.
- Sheldon
- Sheldon
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#8
(11/10/2013, 11:10 PM)sheldonison Wrote:
(11/09/2013, 07:36 AM)mike3 Wrote: .... The case 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, ) 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 .
I'm also interested in any results for , with pseudo period 2, as per our earlier discussion on this forum. I haven't gotten any further with that base, though I believe it has a solution possible due to results I've gotten for a base with pseudo period=5, via a cumbersome indirect method.
- Sheldon

Hmm. Well I got it to work for base , which I was not able to do via the Kneser method. (I can post a graph to show you what looks like, if you want.) I tried it for , but the problem there is that the initial guesses I use for the method are not good enough in that they do not wrap the right way around 0, which matters because log is multivalued.

There are apparently methods by which the initial approximation for the HAM (and also, its other parameters) can be constructed, but I'll have to consult with the local university's library to get the paper on how to do it. The approximation I am using right now is apparently not good enough to do base as it wraps the wrong way around 0. So you may have to wait some, though I'm fiddling with it right now so maybe I might get something.

Added: I've tried forcing the initial guess to wrap the other way, but apparently it isn't close enough to the true solution so that, as iteration proceeds, they jump back across 0 and the method fails. I'm giving up on fiddling with it for now. I'll have to get that paper and see what information it contains.
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#9
(11/10/2013, 05:16 PM)sheldonison Wrote:
(11/09/2013, 07:36 AM)mike3 Wrote: 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 and a lot of complex bases, it doesn't seem to work for the difficult challenge bases , , and .....
I'm on my cellphone... not computer. This method sounds very exciting! You should publish it. Biggest problem with Kouznetsov's method is finite rectangle in imag (z) and discreet sampling. Perhaps an infinite rectangle ?Riemann mapping? to a unit circle? Probably not the approach you're thinking about ...

Anyway, I would definitely be interested in details on your new ideas, and look forward to subsequent posts. Does it work for real bases less than ? Kouznetsov's method relies on limiting behavior at , whereas these bases are periodic in .
- Sheldon

Thought I'd comment on the idea for real bases less than . I suspect it could, but haven't tried. If we wanted to try and recover the regular iteration for these bases, one could modify the Cauchy integral equation so as to have the upper and lower part of the contour as parallel to the real axis and at distance each equal to the (magnitude of the) period. I suspect then one needs two grids of sample points, one on the imaginary axis and one on the real axis, to achieve the integration, so this would require a significant modification to the existing program.

Kouznetsov mentioned about this here:
http://math.eretrandre.org/tetrationforu...d-248.html

On the other hand, it may also be possible to generate the merged Kneser solution (complex-valued at the real axis but (apparently) analytically compatible with the solution for ) using a modified Cauchy integral equation where the axis the contour envelope is skew, going diagonally from the lower left part of the plane to the upper right, on which the function would behave as one asymptotically approaching the fixed points, like for other bases. This would require less modification, since we can still make do with only one grid.

Such a contour would look something like this:
(the graph in the background is for base , obtained via your method, and the dotted line is the one on which the sampling nodes would be put)

   
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#10
This is where I'm hung up on further progress with this method: The HAM method requires the choice of an "auxiliary linear operator". This is a free parameter in the method equations.

The rule, it seems, for choosing this linear operator is to make it one which annihilates some (apparently, 3) initial terms of a "solution expression", which is a form in which to write the solution of the integral or differential equation in question. What we need is to find a form for the tetrational (or, perhaps, better, the solution of the Cauchy equations, which are given in an approximate form so as to approximate it) which looks like



where are properly-chosen "basis functions". I have thought about the Kneser-mapping solution (i.e. regular iteration warped with theta mapping) as a possible set of basis functions, which gives a double summation over coefficients with terms (this is for base . is the fixed point of the logarithm) (note the sum over two indices instead of one), but the problem is this only covers half of the plane (as given, the upper half-plane), and the Cauchy equations require both halves of the plane.

Do you, perhaps, have any ideas as to how this could be done? The form of solution need not converge on the entire plane, only on and perhaps near the imaginary axis.
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