My Math Forum Richardson extrapolation

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 December 12th, 2018, 01:26 PM #1 Member   Joined: Feb 2018 From: Canada Posts: 46 Thanks: 2 Richardson extrapolation Consider a regular n-sided polygon tangential to and enclosing the unit circle to approximate $\pi$. a) Use geometrical arguments to show that the half-perimeter of the polygon $C_n=n\tan(\pi /n) \geq \pi$. b) Calculate $C_4$ and $C_8$. c) Use Richardson extrapolation with $h = 1/n^2$ to improve upon your results. For part b) I've got: $C_4 = 4\tan(\frac{\pi}{4}) = 4\times 1 = 4,$ $C_8 = 8\tan(\frac{\pi}{8}) = 8 \frac{\sin(\frac{\pi}{8})}{\cos(\frac{\pi}{8})} = 8\frac{\frac{\sqrt{2-\sqrt{2}}}{2}}{\frac{\sqrt{2+\sqrt{2}}}{2}} = 8\frac{\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}$ By drawing, I have been able to do the part a) but I have no clue how to do part c). Can someone provide me some help? Thank you. Last edited by skipjack; December 12th, 2018 at 07:00 PM.
 December 12th, 2018, 01:29 PM #2 Senior Member   Joined: Aug 2012 Posts: 2,355 Thanks: 737 Today I learned. https://en.wikipedia.org/wiki/Richardson_extrapolation Thanks from topsquark and Shanonhaliwell
 December 12th, 2018, 07:04 PM #3 Global Moderator   Joined: Dec 2006 Posts: 20,921 Thanks: 2203 tan($\pi$/8) = √2 - 1

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