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A regular n sided polygon inscribed in a circle of diameter one has the circumference as

cn = 2 n Sin( Π / n )

put h=1/n

How can i show whether c(h) = c1/h satisfies the assumption for repeated Richardson extrapolation

Also i need to calculate c2 , c3 , c6 (which i can do ) and perform repeated Richardson Extrapolation

My knowledge in Richardson Extrapolation is very bad and i have got no idea how to do it

So please help

Roushan Singh
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  • The question in the link do appear to be like one given here but only in terms of what is given. But the core question asked is very diffferent. I request you guys to look in the problem again and remove the duplicate mark from it – Roushan Singh Apr 04 '19 at 15:49
  • The accepted answer to the linked question addresses your question as well. It shows how to derive a new asymptotic error expansion from the original. This is the information you need to justify repeated Richardson extrapolation. If your question is about how to combine specific values $c_j$ for $j=2,3,6$ to estimate $\pi$, then an answer is possible, but will not involve what is traditionally labeled as repeated Richardson extrapolation. – Carl Christian Apr 04 '19 at 20:40

1 Answers1

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Just use Taylor's formula: $$ C(h)=\frac 2h \sin (\pi h) = \frac 2h \left((\pi h)- \frac{(\pi h)^3}{3!} + O(h^5) \right) = 2 \pi + \frac{2 \pi^3}{3!} h^2 + O(h^4) $$

In general, when you are approximating a quantity $C^*$ by a formula like $$ C^* = C(h) + a_0 h^{k_0} + O(h^{k_1}), $$

the extrapolation based on $C(h)$ and $C(h/t)$ is given by $$ R(h,t)=\dfrac{t^{k_0} C(h/t)-C(h)}{t^{k_0}-1} $$

and in fact $$ C^* = R(h,t)+O(h^{k_1}) $$

In your case, the extrapolation based on $c_3, c_6$ is given by $$ C^* \approx \dfrac{2^2 c_6-c_3}{2^2-1}=\frac 43 c_6-\frac 13 c_3 $$

You can see the advantage of this procedure just by computing the errors: $$ |2 \pi -C(1/3)|\approx 1.09, \quad |2 \pi -C(1/6)|\approx 0.28, \quad |2\pi - R(1/3,2)| \approx 0.015 $$

PierreCarre
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