R - Richardson Extrapolation for Accurate Estimates

[blackthunder]

Hey, I was hoping someone could help me with this question I can't get at all.

If $$\phi{h}=L-c_1h^{\frac{1}{2}}-c_2h^{\frac{2}{2}}-c_3h^{\frac{3}{2}}-...$$ , then what combination of $$\phi{h}$$ and $$\phi(\frac{h}{2})$$ should give a more accurate estimate of L.

Thanks for any help.

 

[CaptainBlack]

Hey, I was hoping someone could help me with this question I can't get at all.

If $$\phi{h}=L-c_1h^{\frac{1}{2}}-c_2h^{\frac{2}{2}}-c_3h^{\frac{3}{2}}-...$$ , then what combination of $$\phi{h}$$ and $$\phi(\frac{h}{2})$$ should give a more accurate estimate of L.

Thanks for any help.

Assuming that the \(c_i\)s are unknown we can write:

\[\phi( h)=L-c_1h^{1/2}+O( h)\]

and \(n=1/2\) in the Richardson extrapolation formula and so the Richardson extrapolation for \(L\) is:

\[R_L=\frac{2^{1/2}\phi(h/2)-\phi( h)}{2^{1/2}-1}\]

CB