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 Comparing the Known Tetration Solutions jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 08/29/2007, 12:06 AM (This post was last modified: 08/29/2007, 01:07 AM by jaydfox.) I finally got around to comparing my solution with Andrew's. Here's a graph of $S(T(x))$, where $S(x)$ is Andrew's slog for base e, and $T(x)$ is my tetration solution for base e:     At first blush, it looks like we're giving the same results. However, if we look at a graph of $S(T(x))-x$, we can see the discrepancies:     As you can see, the peak error occurs near x=0.54+k, k an integer, and it peaks at about 0.00078 or so. That's an error on the input to Andrew's $\text{slog}^{\small -1}(x)$ function, so it gets magnified on the output as we move away from the critical interval. Since the function is essentially linear on this interval (to within a few percent), we can basically say that the error between our two solutions is about 0.1% or less on the critical interval. My main interest now is to figure out if that cyclic function is indeed a simple sine wave, or if it has a more complex structure. If it's a pure sine wave, and if we can deduce the amplitude and offset, then we could use my solution (which can easily generate hundreds of digits of precision) to calculate Andrew's. I suspect it isn't a pure sine wave, because that would make this just too easy. At any rate, a difficulty here is that I can only estimate the amplitude and offset based on solutions to relatively small systems with Andrew's method. I say "relatively" small, because 560 terms seems like a lot (it took 10.5 hours in SAGE, which seems to be using the maxima engine), and yet given the convergence behavior, I still don't have enough information to understand it. I would need a much larger solution, possibly a system with thousands of terms, and that moves us into supercomputer territory. Any chance we can convince someone with a supercomputer to calculate a relatively large system, say 2000x2000? ~ Jay Daniel Fox « Next Oldest | Next Newest »

 Messages In This Thread Comparing the Known Tetration Solutions - by bo198214 - 08/17/2007, 09:13 PM RE: Comparing the Known Tetration Solutions - by jaydfox - 08/20/2007, 09:59 AM RE: Comparing the Known Tetration Solutions - by bo198214 - 08/20/2007, 11:09 AM New pictures from the hyperbolic slog! - by bo198214 - 08/20/2007, 01:36 PM RE: Comparing the Known Tetration Solutions - by jaydfox - 08/20/2007, 05:35 PM RE: Comparing the Known Tetration Solutions - by jaydfox - 08/21/2007, 05:06 AM RE: Comparing the Known Tetration Solutions - by andydude - 08/22/2007, 11:28 PM RE: Comparing the Known Tetration Solutions - by bo198214 - 08/23/2007, 12:14 AM RE: Comparing the Known Tetration Solutions - by Gottfried - 08/29/2007, 06:19 AM RE: Comparing the Known Tetration Solutions - by jaydfox - 08/29/2007, 06:15 PM RE: Comparing the Known Tetration Solutions - by jaydfox - 08/29/2007, 12:06 AM RE: Comparing the Known Tetration Solutions - by jaydfox - 08/29/2007, 01:56 AM RE: Comparing the Known Tetration Solutions - by bo198214 - 08/29/2007, 07:48 AM RE: Comparing the Known Tetration Solutions - by andydude - 08/29/2007, 08:19 AM RE: computing the iterated exp(x)-1 - by andydude - 08/17/2007, 11:20 PM RE: computing the iterated exp(x)-1 - by jaydfox - 08/17/2007, 11:38 PM RE: computing the iterated exp(x)-1 - by bo198214 - 08/17/2007, 11:45 PM RE: computing the iterated exp(x)-1 - by jaydfox - 08/18/2007, 12:19 AM RE: computing the iterated exp(x)-1 - by bo198214 - 08/18/2007, 08:19 AM RE: computing the iterated exp(x)-1 - by andydude - 08/18/2007, 09:35 AM RE: computing the iterated exp(x)-1 - by bo198214 - 08/18/2007, 11:59 AM RE: computing the iterated exp(x)-1 - by jaydfox - 08/18/2007, 03:49 PM RE: computing the iterated exp(x)-1 - by jaydfox - 08/19/2007, 12:50 AM RE: Comparing the Known Tetration Solutions - by bo198214 - 08/19/2007, 10:55 AM

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