Richardson Extrapolation

Concept

Combing approximation function with different step to reduce the most significant error item.

Example

$\tilde{F}(h)$ is approximation to $F$ where $\tilde{F}(h)=F+c_1{h}^7+O(h^9)$ and we want to reduce the $h^7$ error term.
We can use $\hat{F}(h)=\alpha \tilde{F}(h)+\beta \tilde{F}(2h)$ as new approximation where

• $\alpha +\beta =1$ hold, because we need at least $\hat{F}(h)\approx F$ hold to keep accuracy, other wise it will has term with $(1+\epsilon )F$
• $c_1(\alpha h^7+\beta (2h{)}^7)=0$ to reduce the term