Definition:Richardson Extrapolation

Definition

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 $\map f {\dfrac h 2}$.

Richardson extrapolation is also known as deferred correction.

This entry was named for Lewis Fry Richardson.