A comparison of some of my favorite computational methods for estimating the ratio of a circle's circumference to its diameter.
mypi uses 20 terms or iterations of various computational methods to estimate pi to demonstrate their convergence properties and prints the results to the screen.
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Grid Area By creating an N-gridded square of side length 2 centered at the origin and computing each grid vertex's distance from the origin, we know the total number of vertices that are inside the area of the circle, and we can do a Riemann?Stieltjes integral approximation to the area of the circle using the constructed grid size.
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Monte Carlo (MC) Area By selecting N random points inside [0,1]x[0,1] and computing their distance to the origin, we can find the fraction of points that are inside that quarter of the unit circle. This four times this fraction gives us an estimate of the area of the circle, which, as constructed, is pi.
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Perimeter By creating a 2N-gon all of whose vertices are along the circumference of a circle of radius one, we can compute exactly its perimiter which will approximate that of the circumscribing circle, which is 2pi.
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Zeta(2) Zeta(s) is given by:
inf
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\ 1 /
zeta(s) = / / n^s
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n = 1
Zeta(2) = pi^2/6, so we can truncate this series at n = N to approximate.
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Zeta(4) = pi^4/90, so we can truncate the series at n = N to approximate.
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Zeta(10) = pi^10/93555, so we can truncate the series at n = N to apprixmate.
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Generalized Continued Fraction (GCF) [3 + ...] & [4 / ...] These are implementations of the patterned generalized continued fractions provided in [1].