As you can see, the root test would appear to have a very well-behaved asymptotic root test, right from the start. Excepting the first 10 terms or so, it appears to be approaching an asymptote of about 0.729 or 0.730. This can be seen by zooming in even further on the root test of the derivative:
It's a bit messy, but I hope you can see my point. Presumably, with a significantly larger system (5000x5000, etc.), this should resolve itself into a very tidy pattern. For now, the best I can do is try to pick out the patterns.
One simple test I've found informative is to graph the terms of the derivative, scaled by a factor appropriate to undo the exponential decrease in size that's visible from the root test. In SAGE (notice I load a precalculated set of coefficients, which I can comment out on successive runs to save time):
The resulting graph:
It's a bit messy, but I hope you can see my point. Presumably, with a significantly larger system (5000x5000, etc.), this should resolve itself into a very tidy pattern. For now, the best I can do is try to pick out the patterns.
One simple test I've found informative is to graph the terms of the derivative, scaled by a factor appropriate to undo the exponential decrease in size that's visible from the root test. In SAGE (notice I load a precalculated set of coefficients, which I can comment out on successive runs to save time):
Code:
Flt = RealField(1024)
FltScale = Flt(1.374)
#cofs560 = load('coeffs560.sobj')
sle_coeffs = vector(Flt, len(cofs560), cofs560)
rad1 = [[k-1, Flt(((-1)^(Integer((k-1)/2))*k*sle_coeffs[k])*(FltScale^(k-1)))] for k in range(1, len(sle_coeffs), 2)]
rad2 = [[k-1, Flt(((-1)^(Integer((k-2)/2))*k*sle_coeffs[k])*(FltScale^(k-1)))] for k in range(2, len(sle_coeffs), 2)]
L560_1 = list_plot(rad1, plotjoined=true, rgbcolor=(0.00, 0.00, 0.85))
L560_2 = list_plot(rad2, plotjoined=true, rgbcolor=(0.00, 0.50, 0.85))
G = Graphics()
G+=L560_1
G+=L560_2
G.save('dev-cycles-560.png', xmin=0, xmax=300, ymin=-1.0, ymax=1.0, figsize=(7,5))The resulting graph:
~ Jay Daniel Fox