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?
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