MHB R - Richardson Extrapolation for Accurate Estimates

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To achieve a more accurate estimate of L using Richardson extrapolation, the formula \(R_L=\frac{2^{1/2}\phi(h/2)-\phi(h)}{2^{1/2}-1}\) can be applied. This approach utilizes the values of \(\phi(h)\) and \(\phi(h/2)\) to minimize error terms associated with the unknown coefficients \(c_i\). The extrapolation effectively reduces the leading error term, enhancing the precision of the estimate. The discussion emphasizes the importance of selecting the right combination of these values for improved accuracy. Overall, applying Richardson extrapolation is a key strategy for refining estimates in numerical analysis.
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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.
 
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blackthunder said:
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
 
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