I finally got around to comparing my solution with Andrew's. Here's a graph of , where is Andrew's slog for base e, and 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 , 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 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 , 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 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