Category:Richardson Extrapolation
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This category contains results about Richardson Extrapolation.
Definitions specific to this category can be found in Definitions/Richardson Extrapolation.
Let $\map f h$ be an approximation to an unknown quantity $a$ of the form:
- $\map f h = a + c_1 h^2 + c_2 h^4 + \cdots$
where $h$ is small and $c_1$, $c_2$ and so on are unknown constants.
Richardson extrapolation forms the new approximation to $a$:
- $\hat f = \dfrac 1 3 \paren {\map f h - 4 \map f {\dfrac h 2} }^4$
Hence from:
\(\text {(1)}: \quad\) | \(\ds \map f h\) | \(=\) | \(\ds a + c_1 h^2 + c_2 h^4 + \cdots\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds \map f {\dfrac h 2}\) | \(=\) | \(\ds a + c_1 \dfrac {h^2} 4 + c_2 \paren {\dfrac h 2}^4 + \cdots\) |
it can be seen that the new approximation is obtained by subtracting $4$ times equation $(2)$ from equation $(1)$ to eliminate the $h^2$ terms.
Hence $\hat f$ can be expected to be a more accurate approximation to $a$ than either $\map f h$ or $\map f {\dfrac h 2}$.
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