11/28/2013, 06:53 AM
I've been experimenting with some other functions and mappings for this purpose. There do exist more complicated mappings which map the unit circle to the imaginary axis twice, with strong divergence to imaginary infinity at their singularities, so that the wrapped function will more quickly approach the fixed point and so be smoother. One such function is:
\( f(z) = \tanh\left(\frac{\pi}{4} (x - 1/x)\right) \)
(unit circle -> imag axis).
The inverse, giving the basis functions as powers of this, is
\( b(z) = f^{-1}(z) = \frac{1}{2} \left( \frac{4}{\pi} \tanh^{-1}(z) + \sqrt{\frac{16}{\pi^2} \tanh^{-1}(z)^2 + 4} \right) \).
And the basis functions are \( b_n(z) = b(z)^n \).
The resulting mapping of the tetrational to base \( e \), \( \mathrm{tet}(f(e^{it})) \), looks like (red real, green imag):
We can see this is much smoother and has no nasty corners. When we use this to get the Fourier series, however, we find that both positive and negative degree terms are required (i.e. the sum in terms of \( b_n(z) \) I gave must go from \( -\infty \) to \( \infty \) instead of from \( 0 \) to \( \infty \)), but it is significantly more accurate than the original mapping. In particular, with 203 terms (that's 101 positive and 101 negative degree terms plus the constant term), I get an error on the order of somewhat more than 10^-11 for the tetrational for imaginary-axis inputs close to 0. With 461 terms (230 positive, 230 negative), I get accuracy of around 10^-18 for close to 0, 10^-17 for inputs around \( 20i \). Clearly, this is very much improved, however I'm left wondering if it's possible a still better set of basis functions exists.
While this is no use for getting an analytic approximation out of the HAM since it cannot be integrated exactly (the HAM was originally conceived, actually, as a method to get analytic, i.e. as formulas, approximations to the solutions of nonlinear problems, even though it can be operated as a numerical algorithm as well), we are interested in numerical approximations, and so this should still suffice for that. But it would be interesting if a simple enough basis function existed, that enabled the obtaining of an analytic approximation, since it might help in the finding of a series formula for tetration.
\( f(z) = \tanh\left(\frac{\pi}{4} (x - 1/x)\right) \)
(unit circle -> imag axis).
The inverse, giving the basis functions as powers of this, is
\( b(z) = f^{-1}(z) = \frac{1}{2} \left( \frac{4}{\pi} \tanh^{-1}(z) + \sqrt{\frac{16}{\pi^2} \tanh^{-1}(z)^2 + 4} \right) \).
And the basis functions are \( b_n(z) = b(z)^n \).
The resulting mapping of the tetrational to base \( e \), \( \mathrm{tet}(f(e^{it})) \), looks like (red real, green imag):
We can see this is much smoother and has no nasty corners. When we use this to get the Fourier series, however, we find that both positive and negative degree terms are required (i.e. the sum in terms of \( b_n(z) \) I gave must go from \( -\infty \) to \( \infty \) instead of from \( 0 \) to \( \infty \)), but it is significantly more accurate than the original mapping. In particular, with 203 terms (that's 101 positive and 101 negative degree terms plus the constant term), I get an error on the order of somewhat more than 10^-11 for the tetrational for imaginary-axis inputs close to 0. With 461 terms (230 positive, 230 negative), I get accuracy of around 10^-18 for close to 0, 10^-17 for inputs around \( 20i \). Clearly, this is very much improved, however I'm left wondering if it's possible a still better set of basis functions exists.
While this is no use for getting an analytic approximation out of the HAM since it cannot be integrated exactly (the HAM was originally conceived, actually, as a method to get analytic, i.e. as formulas, approximations to the solutions of nonlinear problems, even though it can be operated as a numerical algorithm as well), we are interested in numerical approximations, and so this should still suffice for that. But it would be interesting if a simple enough basis function existed, that enabled the obtaining of an analytic approximation, since it might help in the finding of a series formula for tetration.