12/18/2007, 12:14 AM

I have finally gotten around to analyzing the natural solutions. The 1024x1024 matrix solution took about 4 hours, with base_ring=RealField(2560) for the curious.

First I show the cumulative absolute error in the coefficients, which sets a conservative upper bound on error in the unit disk. The lines are the same as before, with red lines being solutions of systems with size being an integral power of 2. The blue lines are the half-powers, rounded to the nearest multiple of 8. Notice that there is an extra pair of lines, compared to the accelerated solutions I analyzed. They represent system sizes of 728 and 1024.

Next I show a plot of log(error) vs. log(size). The logarithms are binary logarithms, giving error in bits versus system size in bits.

And there you have it. The natural systems get about 3.5 bits of accuracy per doubling, roughly half that of the accelerated solutions. Therefore, I now feel confident in my original conjecture that my 1200x1200 accelerated solution is more accurate than a 1,000,000x1,000,000 natural solution.

First I show the cumulative absolute error in the coefficients, which sets a conservative upper bound on error in the unit disk. The lines are the same as before, with red lines being solutions of systems with size being an integral power of 2. The blue lines are the half-powers, rounded to the nearest multiple of 8. Notice that there is an extra pair of lines, compared to the accelerated solutions I analyzed. They represent system sizes of 728 and 1024.

Next I show a plot of log(error) vs. log(size). The logarithms are binary logarithms, giving error in bits versus system size in bits.

And there you have it. The natural systems get about 3.5 bits of accuracy per doubling, roughly half that of the accelerated solutions. Therefore, I now feel confident in my original conjecture that my 1200x1200 accelerated solution is more accurate than a 1,000,000x1,000,000 natural solution.

~ Jay Daniel Fox