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Comparing the Known Tetration Solutions - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Comparing the Known Tetration Solutions (/showthread.php?tid=29) |
Comparing the Known Tetration Solutions - bo198214 - 08/17/2007 What really would interest me is a numerical comparison (graphing) of your solution, of Daniel's solution (development at the fixed point of But at least for the first two we have all numerical methods at hand. It simply needs a volunteer (and in the moment my time becomes rare.) ... RE: computing the iterated exp(x)-1 - andydude - 08/17/2007 If by my solution you mean Tetration can be divided into several intervals in terms of what values of x in ![]() Andrew Robbins RE: computing the iterated exp(x)-1 - jaydfox - 08/17/2007 I've gotten SAGE working and I'm currently writing a tetration package for an arbitrary base greater than eta (though it's most stable for bases in a smaller range, probably b<20 or so, depending on precision). I'm using Andrew's SAGE code above to generate the FMA or whatever you would call it, but the rest of the code is based on my cheta function and change of base formula. I'm about halfway done. RE: computing the iterated exp(x)-1 - bo198214 - 08/17/2007 andydude Wrote:If by my solution you meanNo, I meant the piecewise infinite differentiable definition of slog. Isnt it defined for arbirtrary bases? (I think you wrote for base greater 1.) So if you have the slog you have also the "sexp". And I would compare this with what comes out from Daniel's solution for the hyperbolic case (there is then no problem with the convergence radius of the series). Quote:and by Daniel's you meanRather RE: computing the iterated exp(x)-1 - jaydfox - 08/18/2007 bo198214 Wrote:No, the slog function is only defined for bases greater than 1. For bases between 0 and 1, the tetration function is not one-to-one, so its inverse is not a function, because there are multiple values. It's much like saying that y=x^2 does not have an inverse function. y=sqrt(x) only covers part of the domain of the original function.andydude Wrote:If by my solution you meanNo, I meant the piecewise infinite differentiable definition of slog. Also, there's a question about whether you consider the slog function to only apply to the inverse of the function of iterated exponentials/logarithms from 1. For b=2, for example, the domain of slog is negative infinity to 2. However, you can perform iterated exponentials/logarithms from any real number as a starting point, so you could also include the graph for x>4 and the corridor between 2 and 4. RE: computing the iterated exp(x)-1 - bo198214 - 08/18/2007 jaydfox Wrote:No, the slog function is only defined for bases greater than 1. For bases between 0 and 1, the tetration function is not one-to-one, so its inverse is not a function, because there are multiple values. It's much like saying that y=x^2 does not have an inverse function. y=sqrt(x) only covers part of the domain of the original function.Surely, by "arbitrary" I meant: also for I was referring to andydude Wrote:and I suppose would you could call the elliptic? intervalhowever in his pdf he defined it for bases (Quite ridiculous how misunderstandings reach its maximum in the communication between Jay and me ...) Quote:Also, there's a question about whether you consider the slog function to only apply to the inverse of the function of iterated exponentials/logarithms from 1. For b=2, for example, the domain of slog is negative infinity to 2.Dont understand this, Quote: However, you can perform iterated exponentials/logarithms from any real number as a starting point, so you could also include the graph for x>4 and the corridor between 2 and 4.If you use the fixed point method however for base RE: computing the iterated exp(x)-1 - andydude - 08/18/2007 I just wanted to clarify that my super-logarithm solution only works for When I first discovered the super-logarithm solution, the only singularities in the matrix equation (rather than the series expansion) seemed to be less than 1, this is what led me to the requirement For an abstraction of my super-logarithm method, see http://math.eretrandre.org/tetrationforum/showthread.php?tid=27&pid=181#pid181 Andrew Robbins RE: computing the iterated exp(x)-1 - bo198214 - 08/18/2007 andydude Wrote:I just wanted to clarify that my super-logarithm solution only works forAre you sure, what goes wrong for bases I mean it is clear that the domain of the superlog is bounded above by the lower fixed point of So What goes wrong for those arguments with your superlog? RE: computing the iterated exp(x)-1 - jaydfox - 08/18/2007 bo198214 Wrote:Argh, sorry, I meantQuote:Also, there's a question about whether you consider the slog function to only apply to the inverse of the function of iterated exponentials/logarithms from 1. For b=2, for example, the domain of slog is negative infinity to 2.Dont understand this, RE: computing the iterated exp(x)-1 - jaydfox - 08/19/2007 The iterated exp(x)-1 seems to work nicely. It takes a minute or two to generate all the helper functions for a 35-term power series, but then I'm able to calculate 505 points for the interval [-248/128, 256/128], step size 1/128, in just over 59 seconds. This was sufficient to produce the following graph of [attachment=6] By the way, I'm using a radius of 0.005, 35 terms. I could take the number of terms out to 50 pretty easily, and still get increased precision with each new term. As it is, precision seems to be about 65-70 decimal digits. By the way, sorry for the extremely generic appearance of the graph. This was my first attempt at plotting in SAGE, so I haven't tried changing the tick spacings on the axes, adding gridlines and labels, etc. I also plan to add graphs of the derivatives (approximated with secants). |