Wednesday, November 6, 2013

Good approximation for Gauss's circle problem

I found this formula as an approximation to Gauss's circle problem that I described on the last blog entry:
A comparison of the output of this above to the actual values:
r= 0:      0 ~ 1
r=1:       2 ~ 5
r=10:     310.45     ~317
r=20:     1251.38   ~1257
r=30:     2820.99   ~2821
r=40:     5019.11   ~5025
r =45:    6353.84   ~6361
(within .1 % on that last one)
I only found one example where it overestimates:
r=46      6639.63   ~6625

proof: Actually I'm not sure why, but this expression is exact:
because r^2-a^2 is the height of the circle in one quadrant as a increases and the floor of that height counts the number of grid points less than that height then you multiply that by 4 and add the points on the axes with the 4*r term  and add one for the point in the very center.
and then I used the fact that the average value before flooring is probably .5 to get the approximation

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