12/20/2009, 02:08 AM
I just managed to get a code going for doing the regular iteration of \( b = 0.04 \) using the limit formula. I thought this would be a good "Test Bed" or "proof of concept" for this continuum sum method.
Namely, I'm testing the summing of \( \mathrm{reg}_{0.04}(z) \) where \( \mathrm{reg}_{0.04}(0) = 0.5 \).
The graph of this function, \( F(x) = \mathrm{reg}_{0.04}(x) \) for \( x \) from 0 to 12, is given below.
Real part:
Imag part:
Note the very large amplitude oscillation spikes in the graph. The captions at the bottom show the apparent maximum complex magnitude of the function. This appears to be an actual characteristic of the function, it is not a numerical error, and the peak amplitude seems to grow at least tetrationally -- the next oscillation is too intense for the computer to handle, presumably due to a lack of sufficient dynamic range in the floating point math (and we're using bignums here, apparently they don't have bignums in the exponent! Aside: could this be an application for a tetrational representation of extremely gigantic numbers?
). This spiking phenomenon is interesting, as it reminds me of the Gibbs phenomenon in the Fourier approximations of the discontinuous square wave, though with unlimited magnitude and possibly nonzero width in the asymptotic.
This is why I mentioned that for only "most" points in the interval, the function converges to the 2-cycle. If we exclude an interval \( [n + 0.4, n + 0.6] \) (this might actually be too big an interval, but it's enough for illustrative purposes) for integers \( n \), we get this graph (painstakingly prepared by hand from output from the Pari/GP grapher, because you can't exclude intervals
I just set the function to +/-3 at those intervals for graphing and then removed the up/down spikes that resulted by hand) when letting x vary from 0 to 30, and this graph shows of the real part only, the imag part decays to zero, so this one better illustrates the square wave behavior.
Interestingly, the convergence to the square wave seems to be more rapid against \( M_{low} \) than \( M_{high} \). Why is this? Using the square-wave approximation with the discontinuity placed at \( n + 1/2 \) (the exact placement in the interval doesn't affect the value), we get an idea of the appearance of the continuum sum \( \sum_{n=0}^{x-1} \mathrm{reg}_{0.04}(n) \) (and using 256 Mueller terms):
Real part:
Imag part:
Note how it seems we could connect through the gaps at the leftmost parts with a smooth curve (I bet it spikes up though in the gaps toward the right though but there, of course, the square wave approximation ceases to be valid, though I'm unsure of the spiking behavior, if any, in the first gap or two). It would seem that in order to use this method for the full tetration \( \mathrm{tet}_{0.04}(x) \), we'll need a way to analytically continue from a limited interval to a bigger one. Mittag-Leffler expansions are one possibility but their convergence is very slow as was demonstrated in the thread on that subject here. Any ideas about this?
Namely, I'm testing the summing of \( \mathrm{reg}_{0.04}(z) \) where \( \mathrm{reg}_{0.04}(0) = 0.5 \).
The graph of this function, \( F(x) = \mathrm{reg}_{0.04}(x) \) for \( x \) from 0 to 12, is given below.
Real part:
Imag part:
Note the very large amplitude oscillation spikes in the graph. The captions at the bottom show the apparent maximum complex magnitude of the function. This appears to be an actual characteristic of the function, it is not a numerical error, and the peak amplitude seems to grow at least tetrationally -- the next oscillation is too intense for the computer to handle, presumably due to a lack of sufficient dynamic range in the floating point math (and we're using bignums here, apparently they don't have bignums in the exponent! Aside: could this be an application for a tetrational representation of extremely gigantic numbers?
). This spiking phenomenon is interesting, as it reminds me of the Gibbs phenomenon in the Fourier approximations of the discontinuous square wave, though with unlimited magnitude and possibly nonzero width in the asymptotic.This is why I mentioned that for only "most" points in the interval, the function converges to the 2-cycle. If we exclude an interval \( [n + 0.4, n + 0.6] \) (this might actually be too big an interval, but it's enough for illustrative purposes) for integers \( n \), we get this graph (painstakingly prepared by hand from output from the Pari/GP grapher, because you can't exclude intervals
I just set the function to +/-3 at those intervals for graphing and then removed the up/down spikes that resulted by hand) when letting x vary from 0 to 30, and this graph shows of the real part only, the imag part decays to zero, so this one better illustrates the square wave behavior.Interestingly, the convergence to the square wave seems to be more rapid against \( M_{low} \) than \( M_{high} \). Why is this? Using the square-wave approximation with the discontinuity placed at \( n + 1/2 \) (the exact placement in the interval doesn't affect the value), we get an idea of the appearance of the continuum sum \( \sum_{n=0}^{x-1} \mathrm{reg}_{0.04}(n) \) (and using 256 Mueller terms):
Real part:
Imag part:
Note how it seems we could connect through the gaps at the leftmost parts with a smooth curve (I bet it spikes up though in the gaps toward the right though but there, of course, the square wave approximation ceases to be valid, though I'm unsure of the spiking behavior, if any, in the first gap or two). It would seem that in order to use this method for the full tetration \( \mathrm{tet}_{0.04}(x) \), we'll need a way to analytically continue from a limited interval to a bigger one. Mittag-Leffler expansions are one possibility but their convergence is very slow as was demonstrated in the thread on that subject here. Any ideas about this?
